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Interacting particle systems and Jacobi style identities

Interacting particle systems and Jacobi style identities jessica.jay@bristol.ac.uk School of Mathematics, Fry We consider the family of nearest neighbour interacting particle systems on Z allowing Building, University of Bristol, 0, 1 or 2 particles at a site. We parametrise a wide subfamily of processes exhibiting Woodland Road, Bristol BS8 1UG, UK product blocking measure and show how this family can be “stood up” in the sense of Balázs and Bowen (Ann Inst H Poincaré Probab Stat 54(1):514–528, 2018). By comparing measures, we prove new three variable Jacobi style identities, related to counting certain generalised Frobenius partitions with a 2-repetition condition. By specialising to specific processes, we produce two variable identities that are shown to relate to Jacobi triple product and various other identities of combinatorial significance. The family of k-exclusion processes for arbitrary k are also considered and are shown to give similar Jacobi style identities relating to counting generalised Frobenius partitions with a k-repetition condition. 1 Introduction The Jacobi triple product is a two variable identity relating an infinite sum with an infinite product: k k 2i 2i−1 2i−1 −1 q z = (1 − q )(1 + q z)(1 + q z ). i=1 k∈Z It is a foundational identity that has many applications in combinatorics and number theory, e.g. the theory of partitions and the theory of modular forms (Jacobi forms). While the Jacobi triple product is a classical identity and can be proved using elementary ad hoc methods, one finds that most modern proofs exhibit it as an equality of traces of operators acting on certain infinite-dimensional vector spaces. For example, algebraically it is well known that the identity manifests itself in the characters of certain infinite- dimensional representations of affine Lie algebras (e.g. it is the denominator identity for (1) sl ). In physics, it is well known as a consequence of the Boson–Fermion correspondence; both sides of the identity represent the same partition function, calculated in two ways by cataloguing fermionic and bosonic states by their energy and charge relative to a ground state (eigenvalues of operators acting on the two infinite-dimensional Fock spaces of states). © The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 0123456789().,–: volV 48 Page 2 of 46 Balázs et al. Res Math Sci (2022) 9:48 In [3], Balázs and Bowen found a probabilistic proof of the Jacobi Triple Product by interpreting it as an equality between the ASEP blocking measure (product Bernoulli) and the corresponding “stood up” AZRP measure (product geometric). The infinite sum arises via normalisation of the ASEP measure. The ASEP-AZRP correspondence can be seen as the probabilistic analogue of the Boson–Fermion correspondence (with the conserved quantity of a state relative to the zero site being the analogue of charge relative to a ground state). The advantage of the algebraic/physical interpretations of the identity is that they can be vastly generalised, e.g. by considering other representations, other Lie algebras, other Fock spaces. Each approach provides more Jacobi style identities, e.g. the Macdonald identities and the the Rogers–Ramanujan identities. These lead to other interesting connections with partitions and modular forms. Given this, it is natural to ask which other interacting particle systems provide Jacobi style identities. Of course not all processes will work, there are restrictions. For example, in order to get Jacobi style identities it seems necessary to find processes on Z that can be stood up and have interesting blocking measures (product or near product). The only such example for 0-1 processes is ASEP (as seen in [3]). In this paper, we parametrise a family of (non-degenerate) 0-1-2 systems on Z with product blocking measure, purely in terms of conditions on the rates. We then show that they are a valid family to consider in the above sense, giving precise details of stationary blocking measures and stood up processes. The corresponding measures then depend on three parameters q, ˜ t and c, and viewing these as variable over the family we prove the −c following three variable Jacobi style identities (letting z = q˜ ): Theorem 1.1 For 0 < q˜ < 1,t ≥ 1 and z > 0 (+1) 2 i 2 2i −1 i−1 −2 2(i−1) 2 S (˜q, t)˜q z = (1 + tzq˜ + z q˜ )(1 + tz q˜ + z q˜ ) even ∈Z i≥1 i 2 2i −1 i−1 −2 2(i−1) + (1 − tzq˜ + z q˜ )(1 − tz q˜ + z q˜ ) i≥1 (+1) 2+1 i 2 2i −1 i−1 −2 2(i−1) 2t S (˜q, t)˜q z = (1 + tzq˜ + z q˜ )(1 + tz q˜ + z q˜ ) odd ∈Z i≥1 i 2 2i −1 i−1 −2 2(i−1) − (1 − tzq˜ + z q˜ )(1 − tz q˜ + z q˜ ), i≥1 The functions S (˜q, t)and S (˜q, t) are normalising factors for measures on the two even odd irreducible components of the stood up process: 2 I{ω ≥1}− I{ω =0} iω + i(ω −1) −i −i −i −i i odd i even i odd i even S (˜q, t) = q˜ t , even ω∈H 2 I{ω ≥1}− I{ω =0} iω + i(ω −1) −i −i −i −i i even i odd i even i odd S (˜q, t) = q˜ t odd ω∈H Here the state space H is the set of sequences (ω ) of non-negative integers with no −i i≥1 two consecutive zeroes and agreeing with the sequence (0, 1, 0, 1, ... , ) far enough to the right. The state space H is similar but agrees with (1, 0, 1, 0, ...) for enough to the right. Balázs et al. Res Math Sci (2022) 9:48 Page 3 of 46 48 The normalising factors S (˜q, t)and S (˜q, t) as written above might look unappeal- even odd ing (even though they do count explicit combinatorial objects). The form presented here has direct motivation from the standing up procedure of the 0-1-2 state particle systems. In its proof, we demonstrate how this form came to life via such probabilistic ideas. We thank an anonymous referee for pointing out that a purely combinatorial proof is also possible, without the use of probability. We thought it is worth seeing the probability connection while proving Theorem 1.1, hence opted to keep our original argument. We also thank the anonymous referee for the challenge of finding natural alternative forms for S and S . We spent considerable effort finding a closed form of these even odd expressions. While we were not successful, an alternative probabilistic interpretation, giving rather different but still not closed formulas, is subject of a forthcoming paper. Nevertheless, numerical computation to high accuracy has led us to conjecture the fol- lowing equalities: Conjecture 1.2 n+i−1 n−i n n 2i (−1) q˜ t i≥1 n≥i 2i−1 i S (˜q, t) =  + even 2m m 2 (1 − q˜ ) (1 − q˜ ) m≥1 m≥1 n+i 2n+1 2m 3 n−i n(n+1) 2i (−1) q˜ t (1 − q˜ ) m≥1 i≥1 n≥i 2i 2i+1 S (˜q, t) =  +  . odd m 2 m 2 (1 − q˜ ) (1 − q˜ ) m≥1 m≥1 The identities in Theorem 1.1 are shown to have combinatorial significance. By adapt- ing the “General Principle” found in Andrews’ book ( [1]), we find that the RHS of these identities relate to generating functions for generalised Frobenius partitions (GFP’s) satis- fying a 2-repetition condition on the rows and a condition on the number of non-repeated entries. The content of these identities is then that the normalising factors S (˜q, t)and even S (˜q, t) are really formal generating functions for such GFP’s. This is not clear from their odd explicit definitions, and we explain this in detail in Sect. 3.3. In Sect. 4, we specialise to specific 0-1-2 systems and recover various two variable identities, also of combinatorial significance. The ASEP(q, 1) process of Redig et al. (found in [4]) is an extension of classical ASEP to allow two particles. It gives identities relating directly to Jacobi triple product. A (3-state) asymmetric particle-antiparticle exclusion process is considered and gives identities relating directly to the square of Jacobi triple product (which is also seen to have a combinatorial interpretation in terms of 2-coloured GFP’s). The 2-exclusion process gives identities relating to GFP’s with only a 2-repetition condition on the rows. Finally, in Sect. 5 we generalise the last example to the entire family of k-exclusion processes on Z. These are not 0-1-2 systems and so we have to adapt our techniques further. However, they are sufficiently nicely behaved to allow product blocking measure and have stood up processes, which we describe in detail. The identities we recover are as follows. Theorem 1.3 For 0 < q < 1,z = 0 and m ∈{0, 1, ... ,k − 1} k(+1) (k) −m k−m k S (q)q z −m ∈Z 48 Page 4 of 46 Balázs et al. Res Math Sci (2022) 9:48 ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ k−1 k k −rm −αr αi α αr α(i−1) −α ⎝ ⎝ ⎠ ⎝ ⎠ ⎠ = ζ ζ q z ζ q z . k k k r=0 i≥1 α=0 α=0 (k) The functions S (q) for m ∈{0, 1, ... ,k − 1} are normalising factors for measures on −m the k irreducible components of the stood up process: iω + i(ω −1) −i −i (k) i∈ /kZ+m i∈kZ+m S (q) = q . −m −m ω∈H −m Here the state space H is the set of sequences (ω ) of non-negative integers that −i i≥1 have no k consecutive zeroes and satisfy ω = I{i ≡ m mod k} for large enough i (in −i e o analogy with H and H ). The “General Principle” once again explains the combinatorial (k) nature of these identities. The content is that the normalising factors S (q) are formal −m generating functions for GFP’s with k-repetition condition in the rows (functions studied in detail in Andrews’ book [1]). Throughout the paper, we discuss all of the above in both probabilistic and combinatorial detail, giving full justification where possible and otherwise outlining the mysteries implied by one approach to the other. 2 Interacting particle systems with product blocking measure We recall the family of particle systems with product blocking measure introduced by Balázs and Bowen in [3]. For possibly infinite integers, −∞ ≤  ≤ 0 ≤ r ≤∞,wedefine :={i :  − 1 < i < r + 1}⊆ Z, and for two other possibly infinite integers −∞ ≤ min max min max ω ≤ 0 <ω ≤∞,wedefine I :={z : ω − 1 < z <ω + 1}⊆ Z.Weconsider interacting particle systems on the state space  ={η ∈ I :(> −∞ or N (η) < ∞)and (r < ∞ or N (η) < ∞)}, where N ,N : I → Z ∪{∞} are defined by h p h ≥0 0 r min max N (η) = (η − ω )and N (η) = (ω − η ) p i h i i= i=1 min max (when ω = 0and ω < ∞ this is just the number of particles to the left and holes 1 (i,j) to right of ). Given a state η ∈ , the state η ∈  obtained by a particle jumping from site i to j is given by η if k = i, j, (i,j) η = η −1if k = i, k ⎪ η +1if k = j. We will only consider processes with nearest neighbour interactions, i.e. the only states (i,i+1) (i+1,i) we can reach from η are η or η . The system then evolves according to Markov generators; in the ‘bulk’, i.e.  ≤ i ≤ r − 1 the generator has the form r−1 bulk (i,i+1) (i+1,i) L ϕ (η) = p(η , η ) ϕ(η ) − ϕ(η) + q(η , η ) ϕ(η ) − ϕ(η) i i+1 i i+1 i= for some cylinder function ϕ :  → R and functions p, q : I → [0, ∞)(theright andleft jump rates, respectively). If > −∞, we consider an open left boundary with boundary (−1,) jump rates p ,q : I → [0, ∞). We introduce the notation η to denote the state reached from η by a particle entering the system through the left boundary and similarly Balázs et al. Res Math Sci (2022) 9:48 Page 5 of 46 48 (,−1) η to be the state where a particle has left through the boundary. So the left boundary generator is of the form (−1,) (,−1) L ϕ (η) = p (η ) ϕ(η ) − ϕ(η) + q (η ) ϕ(η ) − ϕ(η) . Similarly if r < ∞, we consider an open right boundary with boundary jump rates p ,q : r r (r+1,r) I → [0, ∞). We let η denote the state reached from η when a particle enters the (r,r+1) system through the right boundary and similarly η the state where a particle has left through the boundary. So the right boundary generator is of the form r (r,r+1) (r+1,r) L ϕ (η) = p (η ) ϕ(η ) − ϕ(η) + q (η ) ϕ(η ) − ϕ(η) . r r r r In order to be a member of the blocking family, the jump rates of the system must obey the following conditions: min max (B1) For −∞ <ω <ω < ∞ we have that min max max min p(ω , ·) = p(·, ω ) = q(ω , ·) = q(·, ω ) = 0. If > −∞, min max q (ω ) = p (ω ) = 0 and if r < ∞, max min q (ω ) = p (ω ) = 0. r r (B2) The system is attractive, that is, p(·, ·) is non-decreasing in the first variable and non- increasing in the second whilst q(·, ·) is non-increasing in the first variable and non- decreasing in the second. If > −∞, p is non-increasing and q non-decreasing. If r < ∞, p is non-decreasing and q non-increasing. r r (B3) There exist p ,q ∈ R satisfying < p = 1 − q ≤ 1, and functions asym asym asym asym f : I → [0, ∞)and s : I × I → [0, ∞) such that p(y, z) = p · s(y, z + 1) · f (y)and q(y, z) = q · s(y + 1,z) · f (z). asym asym min min max If ω is finite, then f (ω ) = 0, and if ω is finite, we extend the domain of s max max and require that s(ω + 1, ·) = s(·, ω + 1) = 0. (Note that attractivity implies that s is non-increasing in both of its variables and f is non-decreasing). A priori the above definition allows many of the rates to be zero. However, in this paper we will assume that all rates, except those in (B1), are non-zero. Roughly speaking this min allows us to ignore degenerate processes and assume that f (z) > 0if z >ω (which we will do from now on). In general, there are many stationary distributions for systems satisfying the above. In this paper, we will be interested in the following one-parameter family of product stationary blocking measures, written explicitly in terms of p , q and f . asym asym Theorem 2.1 ([3], Theorem 3.1) For each c ∈ R, there is a product stationary blocking measure μ on , given by the marginals (i−c)z asym 1 q asym μ (z) = for i ∈  and z ∈ I, Z f (z)! i 48 Page 6 of 46 Balázs et al. Res Math Sci (2022) 9:48 ⎪ f (y) for z > 0 y=1 1 for z = 0 where Z is the normalising factor and f (z)! := ⎪ for z < 0. ⎪ 0 f (y) y=z+1 −1 Remark It is clear that if f and s satisfy (B3), then so do αf and α s for any α> 0. This scaling simply shifts the value of c in the blocking measure (i−c)z (i−c˜)z p p asym asym 1 q 1 q asym asym c c˜ μ (z) = = = μ (z). i i Z α f (z)! Z f (z)! i i max min max So without loss of generality we can assume that f (y) = 1 when ω and ω min y=ω +1 are finite. As expected, being a member of the blocking family imposes strict constraints on the min jump rates. In particular, for each y, z ∈ I \{ω } we can use (B3) to write f (z) p q(y − 1,z) asym = . f (y) q p(y, z − 1) asym Setting y = z and recalling that p = 1 − q gives asym asym p(y, y − 1) q(y − 1,y) p = and q = . asym asym q(y − 1,y) + p(y, y − 1) q(y − 1,y) + p(y, y − 1) min Note that p > q implies that p(y, y − 1) > q(y − 1,y) for all y ∈ I \{ω }.This, asym asym along with (B2), shows the condition min max (a) p(y, z) > q(z, y) for all y ∈ I \{ω } and z ∈ I \{ω } (i.e. the process is asymmetric with right drift). 1 1 Also by assumption < p is a constant (as is q < ) and so we must have the asym asym 2 2 condition that p(y, y − 1) min (b) = constant > 1 for all y ∈ I \{ω }. q(y − 1,y) min We then see that for y, z ∈ I \{ω } f (z) p(z, z − 1) q(y − 1,z) = . f (y) q(z − 1,z) p(y, z − 1) f (y) f (z) Since · = 1, this gives the condition f (z) f (y) p(z, z − 1)p(y, y − 1)q(y − 1,z)q(z − 1,y) min (c) = 1 for all y, z ∈ I \{ω }. q(z − 1,z)q(y − 1,y)p(y, z − 1)p(z, y − 1) min It is then clear that the function s is uniquely determined as follows, for y, z ∈ I \{ω } p(y, z − 1) q(y − 1,y) + p(y, y − 1) q(y − 1,z) q(y − 1,y) + p(y, y − 1) ( ) ( ) s(y, z) = = . f (y)p(y, y − 1) f (z)q(y − 1,y) To summarise, conditions (a), (b) and (c) are necessary conditions on the rates that are implied by being a member of the blocking family. However, in general a process with rates only satisfying (B1), (B2) and these three conditions is not expected to be a member of the blocking family (the quantities p and q as written above are well defined asym asym and satisfy < p = 1 − q ≤ 1, but it is not clear that the functions f and s should asym asym exist purely from these conditions). Balázs et al. Res Math Sci (2022) 9:48 Page 7 of 46 48 3 General 0-1-2 systems on Z with blocking measure For the choices I ={0, 1} and  = Z, the only member of the blocking family is ASEP, handled in [3]. In this section, we will consider the case of I ={0, 1, 2} and  = Z. In this case, the conditions (B1), (B2), (B3) translate into the following conditions on the rates (using the assumptions and discussions in Sect. 2): (B1) p(0, ·) = p(·, 2) = q(2, ·) = q(·, 0) = 0, (B2) p(2, ·) ≥ p(1, ·) > 0, p(·, 0) ≥ p(·, 1) > 0, q(0, ·) ≥ q(1, ·) > 0, q(·, 2) ≥ q(·, 1) > 0, (B3) For all y, z ∈{0, 1, 2},wehave p(y, z) = p · s(y, z + 1) · f (y)and q(y, z) = q · s(y + 1,z) · f (z) asym asym with p ,q given by asym asym p(1, 0) 1 q(0, 1) 1 p = > q = < asym asym q(0, 1) + p(1, 0) 2 q(0, 1) + p(1, 0) 2 and f, s given by ⎪ t (q(0, 1) + p(1, 0)) if y = z = 1, ⎧ ⎪ ⎪ tp(1,1)(q(0,1)+p(1,0)) ⎪ ⎪ 0if z = 0, if y = 1and z = 2, ⎪ ⎪ p(1,0) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ p(2,0)(q(0,1)+p(1,0)) −1 f (z):= s(y, z):= t if z = 1, if y = 2and z = 1, p(1,0)t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ p(2,1)(q(0,1)+p(1,0)) t if z = 2, if y = z = 2, p(1,0)t 0if y = 3or z = 3, for some t ≥ 1 (recall that we can assume f (1)f (2) = 1 without loss of generality). f (2) p(1,0)q(0,2) p(1,0)q(0,2) Remark Note that since = t = ,wesee that t = ≥ 1is f (1) q(0,1)p(1,1) q(0,1)p(1,1) uniquely determined. As previously mentioned, (B1), (B2) and (B3) imply the following necessary conditions for the rates: (a) p(y, z) > q(z, y) for all y ∈{1, 2} and z ∈{0, 1}, p(1,0) p(2,1) (b) = , q(0,1) q(1,2) p(1,0)p(2,1)q(1,1)q(0,2) (c) = 1. q(0,1)q(1,2)p(2,0)p(1,1) 48 Page 8 of 46 Balázs et al. Res Math Sci (2022) 9:48 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 e o (a) The “even” ground state, η (b) The “odd” ground state, η e o Fig. 1 Two ground states, η and η ,of Unlike the general case, these conditions are now also sufficient, in the sense that any process with I ={0, 1, 2} and  = Z satisfying (B1), (B2) and conditions (a), (b)and(c)will automatically satisfy (B3) with p ,q ,f and s as given above (all well defined). Thus, asym asym we have fully parametrised the family of such blocking processes using only conditions on the rates. By Theorem 2.1, for each such process there is a one parameter family of product stationary blocking measures μ given by the marginals I{z=1} −(i−c)z t q˜ μ (z) = for z ∈{0, 1, 2}. i c Z (˜q, t) asym q(0,1) c −(i−c) −2(i−c) Here 0 < q˜ = = < 1and Z (˜q, t):= 1 + tq˜ + q˜ (the normalising p i p(1,0) asym factor). Explicitly: ∞ 0 ∞ I{η =1} −(i−c)η I{η =1} (2−η )(i−c) i i i i t q˜ t q˜ c c μ (η) = μ (η ) = . i c c 2(i−c) Z (˜q, t) q˜ Z (˜q, t) i i i=−∞ i=−∞ i=1 Remark Note that these measures only depend on the parameter c and the quantities q, ˜ t attached to the process. Roughly speaking this will be the reason for getting a three variable identity later; as we run through the whole family of such processes, these parameters will be variables. Specialising to particular subfamilies of processes, for example, fixing t or letting t and q˜ be related will give two variable identities (c will still be a variable as will q˜, since the rates will typically depend on an asymmetry parameter 0 < q < 1). By definition of the state space, each η ∈  has η = 0 for all i small enough and η = 2 i i for all i big enough. We refer to these events as having a left most particle (LMP) and right most hole (RMH). By asymmetry, it then follows that the ground states of  (i.e. the most probable states) are all shifts of the following “even” and “odd” ground states (as in Fig. 1): 2if i ≥ 1, ⎨ ⎨ 2if i ≥ 1, e o η = η = 1if i = 0, i i ⎩ ⎪ 0 otherwise, ⎪ 0 otherwise. The reason for the terms “odd” and “even” will become clear in the next section. 3.1 Ergodic decomposition of ∞ 0 For any η ∈ , the quantity N(η):= (2 − η ) − η (as defined in [3]) is finite i i i=1 i=−∞ and is conserved by the dynamics of the process. So we can decompose  =  , n∈Z into irreducible components  :={η ∈  : N(η) = n}. Note that the left shift operator τ, n n−2 n defined by (τη) = η , gives a bijection  − →  (i.e. if η ∈  , then N(τη) = n − 2). i i+1 Balázs et al. Res Math Sci (2022) 9:48 Page 9 of 46 48 3 5 9 11 1 2 4 6 7 8 10 LMP 0 RMH -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 −1 (A) A state η ∈ Ω . (B) The stood up state ω. Fig. 2 An example of standing up −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 e o Fig. 3 “Even” and “odd” states ω and ω e e Remark Since N(η ) = 0, the shifts of η have even conserved quantity and give the n o ground states for the “even” part  of . Similarly, N(η ) =−1 and shifts provide n∈2Z n e o the ground states for the “odd” part  . This explains the labels η and η . n∈2Z+1 n,c c n We now calculate ν (·):= μ (·|N(·) = n), the unique stationary distribution on  . Lemma 3.1 The following relation holds c 2c−N(η) c μ (τη) = q˜ μ (η). This gives the recursion c n−2c c μ ({N = n}) = q˜ μ ({N = n − 2}). Proof (These are special cases of Lemma 6.1 and Corollary 6.2 in [3].) 0 ∞ I{η =1} −(i−c)η I{η =1} (2−η )(i−c) i+1 i+1  i+1 i+1 t q˜ t q˜ μ (τη) = c c 2(i−c) Z (˜q, t) q˜ Z (˜q, t) i=−∞ i=1 0 ∞ I{η =1} −(j−1−c)η I{η =1} (2−η )(j−1−c) j j j j t q˜ t q˜ j=−∞ j=1 2c = q˜ 2(i−c) q˜ Z (˜q, t) Z (˜q, t) i=1 i=−∞ 0 ∞ I{η =1} −(j−c)η I{η =1} (2−η )(j−c) j j j j 0 ∞ t q˜ t q˜ 2c+ η − (2−η ) j j j=−∞ j=1 j=−∞ j=1 = q˜ 2(i−c) q˜ Z (˜q, t) Z (˜q, t) i=1 i=−∞ 2c−N(η) c = q˜ μ (η). Since N(τη) = N(η) − 2, we have that c c μ ({N = n − 2}) = μ (η) η:N(η)=n−2 = μ (τ ηˆ) ηˆ:N(τ ηˆ)=n−2 2c−N(ˆ η) = q˜ μ (ˆ η) ηˆ:N(ˆ η)=n 48 Page 10 of 46 Balázs et al. Res Math Sci (2022) 9:48 2c−n c = q˜ μ ({N = n}). The general solution to the recursion above is n(n+2) ⎨ −nc c q˜ μ ({N = 0})if n ∈ 2Z, μ ({N = n}) = 2 (n+1) ⎩ −(n+1)c c q˜ μ ({N =−1})if n ∈ 2Z + 1. Since there is a dependence on parity, we will need to calculate the probability μ ({N(η) ∈ n,c 2Z + 1}) in order to finish our calculation of ν . Lemma 3.2 1 − 1 − 2μ (1) i=−∞ μ ({N(η) ∈ 2Z + 1}) = . Proof (We adapt the proof of [5], Proposition 1). Define the partial conserved quantity a 0 N (η) = (2 − η ) − η for a ≥ 1 a i i i=1 i=−a and note that N (η) → N(η)as a →∞. For each a ≥ 1, we define the random variables N (η) i=−a Y := (−1) = (−1) . Since η → 0as i →−∞ and η → 2as i →∞, we have that Y → Y as a →∞, where i i a 1if N(η) is even, N(η) Y := (−1) = −1if N(η) is odd. Since |Y |= 1 for all a ≥ 1, dominated convergence applies, and by the product structure of μ we have that ∞ ∞ c c c c c E [Y ] = lim E [Y ] = 1 · μ (0, 2) + (−1) · μ (1) = 1 − 2μ (1) , i i i a→∞ i=−∞ i=−∞ c c where E denotes the expectation taken w.r.t. μ . On the other hand, we have c c c E [Y ] = 1 · μ {N(η) ∈ 2Z} + (−1) · μ {N(η) ∈ 2Z + 1} = 1 − 2μ {N(η) ∈ 2Z + 1} . Thus, 1 − 1 − 2μ (1) i=−∞ μ ({N(η) ∈ 2Z + 1}) = and also 1 + 1 − 2μ (1) i=−∞ c c μ ({N(η) ∈ 2Z}) = 1 − μ ({N(η) ∈ 2Z + 1}) = . Balázs et al. Res Math Sci (2022) 9:48 Page 11 of 46 48 Now by combining Lemma 3.1 and Lemma 3.2, we get the unique stationary distribution n,c n ν on  . Proposition 3.3 For n ∈ 2Z, the unique stationary distribution on  is given by (+1)−2c c 2 q˜ μ (η)I{N(η) = n} =−∞ n,c ν (η) = . n(n+2) −nc q˜ 1 + 1 − 2μ (1) i=−∞ For n ∈ 2Z + 1,itisgiven by (+1) −2(+1)c c 2 q˜ μ (η)I{N(η) = n} =−∞ n,c ν (η) = . 2 ∞ (n+1) −(n+1)c q˜ 1 − 1 − 2μ (1) i=−∞ −(i−c) −2(i−c) 1−tq˜ +q˜ c c −(i−c) Notice that (1 − 2μ (1)) = , and so if we let W (˜q, t):= 1 − tq˜ + i i Z (˜q,t) −2(i−c) n,c q˜ ,wecan write ν as ∞ 0 ∞ (+1)−2c I{η =1} −η (i−c) I{η =1} (2−η )(i−c) i i i i 2 q˜ t q˜ t q˜ =−∞ i=−∞ i=1 n,c ν (η) = ∞ ∞ n(n+2) −nc c c c c 2(i−c) 2(i−c) q˜ q˜ Z (˜q, t)Z (˜q, t) + q˜ W (˜q, t)W (˜q, t) i −i+1 i −i+1 i=1 i=1 I{N(η) = n} when n is even and ∞ 0 ∞ (+1) −2(+1)c I{η =1} −η (i−c) I{η =1} (2−η )(i−c) i i i i 2 q˜ t q˜ t q˜ =−∞ i=−∞ i=1 n,c ν (η) = ∞ ∞ (n+1) −(n+1)c c c c c 2(i−c) 2(i−c) q˜ q˜ Z (˜q, t)Z (˜q, t) − q˜ W (˜q, t)W (˜q, t) i −i+1 i −i+1 i=1 i=1 I{N(η) = n} when n is odd. Remark These distributions are independent of c, as discussed in [3]. However, later we will need to stress the dependence of both the numerator and denominator on c and so we will keep c in our notation. 3.2 Standing up/laying down In this section, we transfer the dynamics on  to that of a restricted particle system on <0 Z , in direct analogy with the “standing up/laying down” method of [3]. By doing this, we ≥0 obtain an alternative characterisation of the stationary distributions given in Proposition 3.3. n th Definition 3.4 Given η ∈  ,let S be the site of the r particle when reading left to n <0 right, bottom to top. The corresponding stood up state is then T (η) = ω ∈ Z ,with ≥0 ω = S − S .See Fig. 2 for example. −r r+1 r Remark We can extend this method to stand up blocking processes with I = {0, 1, 2, ... ,k} and  = Z for any k ∈ N. However, the corresponding particle systems are 48 Page 12 of 46 Balázs et al. Res Math Sci (2022) 9:48 not always guaranteed to have product stationary blocking measures (this is dependent on the jump rates of the original system). n <0 A priori the “standing up” map T is an injection into Z . However, since η ≤ 2 for all ≥0 i, the image of T lies in the restricted state space <0 H :={ω ∈ Z : ω = 0 ⇒ ω = 0, ∀i > 0}. −i −i−1 ≥0 Since η = 2 for i large ω must coincide far to the left with one of the following “even” or e o e “odd” states (dependent on the parity of n); ω and ω such that ω = I{i ∈ 2Z } and ≥1 −i ω = I{i ∈ 2Z + 1} (see Fig. 3). ≥0 −i e 0 e Remark Note that all shifts of the “even” ground state η ∈  standuptogive ω . o −1 o Similarly all shifts of η ∈  give ω . This is further justification for the notation. n e o e o We now see that the image of T lies in H := H ∪ H , where the disjoint sets H and H are defined as e  e H ={ω ∈ H : ∃N >0s.t ω = ω ∀i ≥ N }, −i −i o  o H ={ω ∈ H : ∃N >0s.t ω = ω ∀i ≥ N }. −i −i For ω ∈ H, the minimum value of N satisfying the above will be denoted by E(ω)or O(ω), e o dependent on whether ω ∈ H or ω ∈ H . ⎨ e H if n ∈ 2Z, n n Lemma 3.5 T ( ) = ⎩ o H if n ∈ 2Z + 1. Proof It suffices to show surjectivity of T for each n. e n If n ∈ 2Z and ω ∈ H , then we construct the state η ∈  having leftmost particle at E(ω)−1 n+E(ω)+I{E(ω)∈2Z+1} th the site S = − ω and r particle at site S = S + ω for 1 −i r r−1 1−r i=1 each r ≥ 2. o n Similarly for n ∈ 2Z + 1and ω ∈ H construct the state η ∈  with left most particle O(ω)−1 n+O(ω)+I{O(ω)∈2Z} th at site S = − ω and r particle at site S = S + ω for 1 −i r r−1 1−r i=1 each r ≥ 2. n n It is clear that in either case T (η) = ω and hence T is surjective. The inverse maps described in the above proof are referred to as the “laying down” maps. Using the “standing up” maps we define a particle system on H whose dynamics are inherited from those on . In particular right jumps in η correspond to right jumps in ω and similarly for left jumps. The explicit right/left jump rates are given in Table 1 for r ≥ 2. Note that the jump rates p (1, ·)and q (·, 1) can be 0, due to the no consecutive zeroes ω ω condition, a jump that would cause ω = ω = 0 for some r ≥ 2 is blocked. −r −r+1 Since the “stood up ” process is only defined on the negative half integer line we must consider what happens at the boundary site. We consider an open infinite type boundary, that is a reservoir of particles at “site 0” at which particles can enter or leave the system with the rates given in Table 2. Balázs et al. Res Math Sci (2022) 9:48 Page 13 of 46 48 Table 1 Jump rates p (ω , ω ) and q (ω , ω ), ω −r −r+1 ω −r −r+1 respectively, of the stood up process ω = 0 ω ≥ 1 −r+1 −r+1 ω =00 0 −r ω = 1 p(2, 1)I{ω = 0} p(1, 1)I{ω = 0} −r −r−1 −r−1 ω ≥ 2 p(2, 0) p(1, 0) −r ω = 0 ω = 1 ω ≥ 2 −r+1 −r+1 −r+1 ω =00 q(1, 2)I{ω = 0} q(0, 2) −r −r+2 ω ≥10 q(1, 1)I{ω = 0} q(0, 1) −r −r+2 Table 2 Boundary jump rates for the stood up process Rate into the boundary Rate out of the boundary ω =00 q(0, 2) −1 ω = 1 p(1, 1)I{ω = 0} q(0, 1) −1 −2 ω ≥ 2 p(1, 0) −1 We note that the dynamics at the boundary in ω correspond exactly to that of the LMP −1 in T (ω). ( ) To find the stationary distribution for the “stood up” process we first consider the ∗ <0 unrestricted process ω ∈ Z , i.e. the process described by the same jump rates as ω ≥0 but where the number of consecutive zeros is not restricted. It is clear to see that the unrestricted process is a member of the blocking family; (B1) and (B2) follow directly from the fact that the rates of the original process satisfy these conditions and similarly condition (B3) is satisfied by the constants p(1, 0) 1 q(0, 1) 1 ∗ ∗ p = > and q = < asym asym q(0, 1) + p(1, 0) 2 q(0, 1) + p(1, 0) 2 and the functions tp(2,1)(q(0,1)+p(1,0)) ⎪ if y = z = 1, ⎪ p(1,0) ⎧ ⎪ ⎪ ⎪ 0if z = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ tp(1,1)(q(0,1)+p(1,0)) ⎪ ⎪ ⎪ ⎪ if y = 1and z ≥ 2, ⎪ ⎪ p(1,0) ⎨ ⎨ ∗ ∗ −1 f (z):= s (y, z):= t if z = 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p(2,0)(q(0,1)+p(1,0)) ⎪ ⎪ ⎪ ⎪ if y ≥ 2and z = 1, ⎪ ⎪ p(1,0)t ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ t if z ≥ 2, ⎩ q(0,1)+p(1,0) if y, z ≥ 2, q(0,1)p(2,0) with t = > 1. p(1,0)q(1,1) By Theorem 2.1 we can find a one parameter family of product stationary blocking ∗,cˆ <0 measures π on Z with marginals given by ≥0 (2−z)I{z≥1} (i+cˆ)z t q˜ ∗,cˆ π (z) = for z ≥ 0and i ≥ 1 −i ∗,cˆ Z (˜q, t) −i ∗,cˆ (here Z (˜q, t) is the normalising factor). −i We fix the value of cˆ by considering reversibility over the boundary edge (−1, 0). Suppose ∗,cˆ that π satisfies detailed balance over this boundary edge: ∗,cˆ ∗,cˆ π (y) · “rate into the boundary” = π (y − 1) · “rate out of the boundary” for all y ≥ 1. −1 −1 We consider the following two cases 48 Page 14 of 46 Balázs et al. Res Math Sci (2022) 9:48 (1) If y = 1, then this gives ∗,cˆ ∗,cˆ π (1)p(1, 1) = π (0)q(0, 2) −1 ∗,cˆ π (1) q(0, 2) p(2, 0)q(0, 1) −1 ⇔ = = by condition (B3)(c) ∗,cˆ 2 p(1, 1) p(1, 0) q(1, 1) π (0) −1 1+cˆ 2 ⇔ tq˜ = t q˜ cˆ ⇔ q˜ = t. (2) If y ≥ 2, then this gives ∗,cˆ ∗,cˆ π (y)p(1, 0) = π (y − 1)q(0, 1) −1 ∗,cˆ π (y) q(0, 1) −1 ⇔ = = q˜ ∗,cˆ p(1, 0) π (y − 1) −1 −1 1+cˆ ⇔ t q˜ = q˜ cˆ ⇔ q˜ = t. cˆ Thus, we should choose cˆ so that q˜ = t. The marginals now become (dropping cˆ from the notation) iz 2I{z≥1} q˜ t π (z) = for z ≥ 0and i ≥ 1. −i ∗ Z (˜q, t) −i Now that we have the stationary distribution for the unrestricted process we consider e o the restriction to H and find the stationary measure. Recall that H = H ∪ H , and note e e o that H is the irreducible component of the “even” ground state ω and similarly H for the “odd” ground state ω . We define stationary measures on these irreducible components ∗ e e o o in terms of π , getting π on H and π on H . It seems natural to define these measures e ∗ e o ∗ o ∗ as π (·) = π (·|· ∈ H )and π (·) = π (·|· ∈ H ). However, w.r.t π the probability of being in either irreducible component is zero and so these quantities are undefined. To rectify this, we use the following formal reasoning, e ∗ e π (ω) = π (ω|ω ∈ H ) ∗ e π (ω )I{ω ∈ H } −i −i i≥1 π (ω ) −i −i i≥1 ω ∈H π (ω ) −i −i I{ω ∈ H } ∗ e π (ω ) −i −i i≥1 = . π (ω ) −i −i ∗ e π (ω ) −i −i ω ∈H i≥1 This is now a well defined distribution since far to the left any configuration ω ∈ H agrees with ω , forcing the products to be finite and the denominator to no longer be 0. We apply a similar reasoning for π . We then have e e o o φ (ω )I{ω ∈ H } φ (ω )I{ω ∈ H } −i −i −i −i i≥1 i≥1 e o π (ω) =   and π (ω) =   , e  o φ (ω ) φ (ω ) −i −i −i −i e  o ω ∈H i≥1 ω ∈H i≥1 Balázs et al. Res Math Sci (2022) 9:48 Page 15 of 46 48 ∗ ∗ π (ω ) π (ω ) −i −i e −i o −i where φ (ω ) = and φ (ω ) = , given explicitly as follows: ∗ e ∗ o −i −i −i −i π (ω ) π (ω ) −i −i −i −i iω 2I{ω ≥1} −i −i q˜ t i ∈ 2Z + 1, φ (ω ):= −i −i i(ω −1) −2I{ω =0} −i −i q˜ t i ∈ 2Z, i(ω −1) −2I{ω =0} −i −i q˜ t i ∈ 2Z + 1, φ (ω ):= −i −i iω 2I{ω ≥1} −i −i q˜ t i ∈ 2Z e o and so we get the following explicit formulae for π (ω)and π (ω). e o Proposition 3.6 The unique stationary measures on H and H are given by 2 I{ω ≥1}− I{ω =0} iω + i(ω −1) −i −i −i −i iodd ieven iodd ieven q˜ t π (ω) = , S (˜q, t) even 2 I{ω ≥1}− I{ω =0} iω + i(ω −1) −i −i −i −i ieven iodd ieven iodd q˜ t π (ω) = . S (˜q, t) odd e o Here S (˜q, t)and S (˜q, t) are normalising factors with respect to H and H : even odd 2 I{ω ≥1}− I{ω =0} iω + i(ω −1) −i −i −i −i i odd i even i odd i even S (˜q, t) = q˜ t , even ω ∈H 2 I{ω ≥1}− I{ω =0} iω + i(ω −1) −i −i −i −i i even i even i odd i odd S (˜q, t) = q˜ t . odd ω ∈H Proof It is well known that the restriction of a reversible stationary measure on a contin- uous time Markov process is also reversible stationary (see Proposition 5.10 of [6]). The o e ∗ result follows since π and π are restrictions of π . It would be interesting to know whether the normalising factors S (˜q, t)and S (˜q, t) even odd can be written as infinite products. We will see later that when specialising to certain processes (i.e. choosing certain values for q˜ and t) the specialised normalising factors appear to be products with combinatorial significance. We should compare the situation with that of [3], in which the authors stand up ASEP to get AZRP and get the partition 2i −1 function (1 − q ) as normalising factor (one of the components of the product i≥1 side of the Jacobi triple product). However, the fact that they naturally get a product seems to be a direct consequence of the choice of “standing up” map. Indeed it is possible to stand up ASEP in a different way, giving a normalising factor that has a similar form to the two given above (i.e. not naturally given as a product) but is then recognised as the above partition function. Unfortunately we do not see an obvious way to recognise the normalising factors S (˜q, t)and S (˜q, t) as products. even odd 48 Page 16 of 46 Balázs et al. Res Math Sci (2022) 9:48 3.3 Identities n n By Lemma 3.5, the standing up transformation T describes a bijection between e o n and one of the state spaces H or H , depending on the parity of n.Since T preserves n,c the dynamics of the corresponding processes, we get an equality of measures, ν (η) = e n n,c o n π (T (η)), when n ∈ 2Z and ν (η) = π (T (η)), when n ∈ 2Z + 1 (for all values of c). We will now see that evaluating these equalities at ground states leads to interesting combinatorial identities. Recall that a process on  with blocking measure has two ground states up to shift, e 0 o −1 0 e e −1 o o 0,c e e e η ∈  and η ∈  , satisfying T (η ) = ω and T (η ) = ω .Thus, ν (η ) = π (ω ) −1,c o o o and ν (η ) = π (ω ), and by Proposition 3.3 and Proposition 3.6 we get the following two identities (after rearrangement): (+1)−2c 2(i−c) c c 2 S (˜q, t)˜q = q˜ Z (˜q, t)Z (˜q, t) even i −i+1 =−∞ i≥1 2(i−c) c c + q˜ W (˜q, t)W (˜q, t), i −i+1 i≥1 (+1) −(2+1)c 2(i−c) c c 2t S (˜q, t)˜q = q˜ Z (˜q, t)Z (˜q, t) odd i −i+1 =−∞ i≥1 2(i−c) c c − q˜ W (˜q, t)W (˜q, t). i −i+1 i≥1 c c −c Writing Z (˜q, t)and W (˜q, t) explicitly and letting z = q˜ prove the following identities. i i Theorem 1.1 For 0 < q˜ < 1,t ≥ 1 and z > 0 (+1) 2 i 2 2i −1 i−1 −2 2(i−1) 2 S (˜q, t)˜q z = (1 + tzq˜ + z q˜ )(1 + tz q˜ + z q˜ ) even i≥1 ∈Z i 2 2i −1 i−1 −2 2(i−1) + (1 − tzq˜ + z q˜ )(1 − tz q˜ + z q˜ ) i≥1 (+1) 2+1 i 2 2i −1 i−1 −2 2(i−1) 2t S (˜q, t)˜q z = (1 + tzq˜ + z q˜ )(1 + tz q˜ + z q˜ ) odd ∈Z i≥1 i 2 2i −1 i−1 −2 2(i−1) − (1 − tzq˜ + z q˜ )(1 − tz q˜ + z q˜ ), i≥1 We were unable to find these three variable Jacobi style identities explicitly written down in the literature, and we believe that they are new. It is interesting to note that the above proof of these identities is purely probabilistic and makes no assumption of other classical identities; for example, Jacobi triple product (although we will see later that specialising to the ASEP(q, 1) process gives the odd/even parts of Jacobi triple product). We will now discuss the combinatorial nature of these identities. As is well known, the product (1 + q˜ ) is the generating function for partitions of n with distinct parts. In i≥1 i 2i a similar vein the product (1 + q˜ + q˜ ) is the generating function for partitions of i≥1 n with each part appearing at most twice. It is then clear that the two variable product i 2i n m (1 + tq˜ + q˜ ) = a q˜ t is the generating function for such partitions of n,m i≥1 n,m n with exactly m distinct parts. For example a = 3and a = 2 count the partitions 5,1 5,2 Balázs et al. Res Math Sci (2022) 9:48 Page 17 of 46 48 764 631 631 764 Fig. 4 Conjugate partitions 30 = 8 + 8 + 7 + 3 + 2 + 1 + 1and 30 = 7 + 5 + 4 + 3 + 3 + 3 + 3 + 2 with their corresponding GFP’s [5, 3 + 1 + 1, 2 + 2 + 1] and [4 + 1, 3 + 2], respectively (the partitions 2 + 1 + 1 + 1and 1 + 1 + 1 + 1 + 1 are not counted since 1 appears more than twice). Going one step further i 2 2i n m k the three variable product (1 + tzq˜ + z q˜ ) = a q˜ t z is the generating n,m,k i≥1 n,m,k function for such partitions of n which have exactly k parts in total. For example a = 5,1,1 1,a = 2and a = 2 count the partitions [5], [3 + 1 + 1, 2 + 2 + 1] and [4 + 1, 3 + 2], 5,1,3 5,2,2 i 2 2i −1 i−1 −2 2(i−1) respectively. The meaning of the product (1 +tzq˜ +z q˜ )(1 +tz q˜ +z q˜ ) i≥1 is more subtle and relates to generalised Frobenius partitions. A generalised Frobenius partition (GFP) of n is a two row array of integers a a ... a 1 2 s b b ... b 1 2 s such that a ≥ a ≥ ··· ≥ a ≥ 0, b ≥ b ≥ ··· ≥ b ≥ 0and s + (a + b ) = n. 1 2 s 1 2 s i i 1≤i≤s Given an ordinary partition of n we can produce a GFP of n by letting s be the length of the leading diagonal in the Young diagram, the a be the lengths of rows to the right of the diagonal and the b be the lengths of the columns under the diagonal (see Fig. 4). This map gives a bijection between ordinary partitions of n and GFP’s of n with each row having distinct entries (note that GFP’s in general allow repeats in the rows). Taking the conjugate of a partition becomes the natural operation of swapping rows in the corresponding GFP. This convenience was the classical motivation, but GFP’s are now studied as combinatorial objects in their own right. Naturally, we wish to count certain families of GFP’s. A “General Principle” due to n k Andrews, given in [1], provides a way to do this. Suppose f (˜q, z) = a q˜ z and n,k n,k n k f (˜q, z) = b q˜ z are generating functions for ordinary partitions of n with k parts n,k n,k and satisfying some conditions A and B, respectively. Then, the “General Principle” states −1 k that the formal series f (˜q, qz ˜ )f (˜q, z ) = f (˜q)z has constant term f (˜q) equal A B A,B,k A,B,0 to the generating function for GFP’s of n with first row satisfying condition A and second row satisfying condition B. The set of such GFP’s will be denoted GFP (n). We can A,B actually give a uniform interpretation of all of the formal series f (˜q) if we generalise A,B,k further to allow GFP’s having rows of unequal length, i.e. two row arrays of integers a a ... a 1 2 s b b ... b 1 2 s such that a ≥ a ≥ ··· ≥ a ≥ 0,b ≥ b ≥ ··· ≥ b ≥ 0and s + a + 1 2 s 1 2 s 1 i 1 2 1≤i≤s b = n. We will refer to these as GFP’s with offset s − s (so that GFP’s with i 1 2 1≤i≤s 2 48 Page 18 of 46 Balázs et al. Res Math Sci (2022) 9:48 offset 0 are classical GFP’s). The |s − s | “empty” entries in the shorter row will be 1 2 labelled to the left with a dash (this will be important later). The full power of the “General Principle” is then that the formal series f (˜q) is the generating function for the sets A,B,k GFP (n), defined as above but for (possibly non-zero) offset k. A popular choice of A,B,k condition on the rows is A = B = D , the condition that each part appears at most r times. One can use the “General Principle” to give a combinatorial interpretation of the famous Jacobi Triple Product identity (written here in an equivalent form to the one given in the introduction): k(k+1) i i i−1 −1 k (1 − q˜ )(1 + q˜ z)(1 + q˜ z ) = q˜ z . i≥1 k∈Z Indeed, rearranging gives: k(k+1) i i−1 −1 k (1 + q˜ z)(1 + q˜ z ) =  q˜ z (1 − q˜ ) i≥1 i≥1 k∈Z and so by applying the “General Principle” to the LHS we see that this identity is equivalent k(k+1) to the equalities f (˜q) = q˜ for all k ∈ Z. This is true for the base case D ,D ,k i 1 1 (1−q˜ ) i≥1 k = 0 since we have already seen that GFP (n) is in bijection with ordinary partitions D ,D ,0 1 1 of n. For other values of k the corresponding equality follows from the equivalence of k(k+1) generating functions f (˜q) = f (˜q)˜q , proved by the following bijection φ : D ,D ,k D ,D ,0 k 1 1 1 1 k(k+1) GFP (n) → GFP n + often attributed to Wright. Given an element D ,D ,0 D ,D ,k 1 1 1 1 2 of GFP (n) we have a corresponding ordinary partition of n. Adjoin a right angled D ,D ,0 1 1 |k|(|k|+1) triangle of size to either the left or top edge of its Young diagram, depending on whether k ≥ 0or k < 0, respectively (see Fig. 5). Then, use the new leading diagonal k(k+1) implied by the triangle to read off an element of GFP n + by letting s D ,D ,k 1 1 2 be the length of the diagonal if k ≥ 0(s if k < 0), the a be the sizes of rows to the 2 i right of the diagonal and the b be the sizes of columns under the diagonal (the |k| empty rows/columns coming from the triangle supply the required |k| “empty” entries in the corresponding GFP of offset k). In comparison, we can now give a combinatorial interpretation of the identities in Theorem 1.1. By a three variable adaptation of the “General Principle” the formal series: i 2 2i −1 i−1 −2 2(i−1) k (1 + tzq˜ + z q˜ )(1 + tz q˜ + z q˜ ) = f (˜q, t)z D ,D ,k 2 2 i≥1 k∈Z n m has coefficients f (˜q, t) = c q˜ t that are two variable generating functions D ,D ,k n,m,k 2 2 n,m for the sets GFP (n) ⊆ GFP (n), consisting of such GFP’s having a total of D ,D ,k,m D ,D ,k 2 2 2 2 m distinct parts (each row treated separately). The identities of Theorem 1.1 are then equivalent to the equalities: ⎨ (+1) S (˜q, t)˜q if k = 2, even f (˜q, t) = D ,D ,k 2 2 2 (+1) tS (˜q, t)˜q if k = 2 + 1. odd (The reason for various sign changes in the two identities is merely to separate the cases of even and odd offset, since these behave differently.) We have of course proved these equalities probabilistically, but it is not explicitly clear that the normalising factors are Balázs et al. Res Math Sci (2022) 9:48 Page 19 of 46 48 −→ 764 11 10 8 3 1 631 −− − − 2 (a) An element of GFP (30) and its image in GFP (40) D ,D ,0 D ,D ,4 1 1 1 1 −4 −→ 764 −− − − 320 631 10 7 5 3 210 (b) An element of GFP (30) and its image in GFP (36) D ,D ,0 D ,D ,−4 1 1 1 1 Fig. 5 Wright bijections φ and φ (cont) 4 −4 related to counting GFP’s with the 2-repetition condition. However, we are able to give combinatorial proofs of these equalities, similar to the case of Jacobi triple product. We start with the base cases k = 0and −1, i.e. that S (˜q, t) = f (˜q, t) even D ,D ,0 2 2 tS (˜q, t) = f (˜q, t). D ,D ,−1 odd 2 2 Using MAGMA, we were able to compute that the first few coefficients of the normalising factors are given by: 2 2 2 3 2 4 2 4 5 2 4 S (˜q, t) = 1 + qt ˜ + q˜ (1 + 2t ) + 5˜q t + q˜ (2 + 6t + t ) + q˜ (12t + 2t ) even 6 2 4 7 2 4 8 2 4 + q˜ (3 + 16t + 5t ) + q˜ (25t + 10t ) + q˜ (5 + 30t + 20t ) + ··· 2 2 3 2 4 2 5 2 S (˜q, t) = 1 + 2˜q + q˜ (2 + t ) + q˜ (4 + 2t ) + q˜ (5 + 5t ) + q˜ (6 + 10t ) odd 6 2 4 7 2 4 8 2 4 + q˜ (10+15t +t )+q˜ (12 + 26t + 2t ) + q˜ (15 + 40t + 5t ) + ··· (see Appendix for the lists of GFP’s of offset 0 and −1 that are counted by these coef- ficients.) Note that in both cases the exponents of t are even. This is expected since − 422 2 200 Fig. 6 Generalised Young diagrams corresponding to two GFP’s of 15 with offset 0 and −1 48 Page 20 of 46 Balázs et al. Res Math Sci (2022) 9:48 GFP (n) is empty when m ≡ k mod 2 (this justifies the extra t in the odd case). D ,D ,k,m 2 2 These expansions will be useful later when specialising to particular processes. In order to prove these base cases we seek analogues of the Frobenius bijection between ordinary partitions of n and elements of GFP (n). However, to describe these we must first D ,D ,0 1 1 generalise the notion of Young diagram to allow GFP’s with the 2-repetition condition. Elements of GFP (n) do not correspond to ordinary partitions and so do not D ,D ,0 2 2 naturally give rise to Young diagrams. However, they do naturally give rise to cer- tain finite subsets of C ={(n ,n ) ∈ Z : n + n ≡ 0 mod 2}. The subset cor- e 1 2 1 2 responding to such a GFP with s columns consists of the s leading diagonal points (1, −1), ... , (s, −s), the first a points of C to the right of (i, −i) and the first b points of C i e i e under (i, −i). Similarly, elements of GFP (n) will give well defined finite subsets of D ,D ,−1 2 2 C ={(n ,n ) ∈ Z : n + n ≡ 1 mod 2}. The subset corresponding to such a GFP with o 1 2 1 2 s entries on the top row contains the s leading diagonal points (2, −1), ... , (s + 1, −s ), 1 1 1 1 the first a points of C to the right of (i + 1, −i) and the first b points of C under i o i o (i, −i + 1) (the point (1, 0) is not included). See Fig. 6 for an example of each kind of generalised Young diagram (points are labelled by black squares; the white squares are included only for aesthetics). n m We can now return to proving the base cases. By comparing coefficients of q˜ t on both sides it suffices to find bijections: iω + i(ω − 1) = n e e −i −i i odd i even ψ : ω ∈ H : → GFP (n), D ,D ,0,m n,m 2 2 2 I{ω ≥ 1}− I{ω = 0} = m −i −i i odd i even iω + i(ω − 1) = n o o −i −i i even i odd ψ : ω ∈ H : → GFP (n). D ,D ,−1,m+1 n,m 2 2 2 I{ω ≥ 1}− I{ω = 0} = m −i −i i even i odd These maps can be constructed as restrictions of bijections: e e ψ : ω ∈ H : iω + i(ω − 1) = n → GFP (n) −i −i D ,D ,0 n 2 2 i odd i even o o ψ : ω ∈ H : iω + i(ω − 1) = n → GFP (n). −i −i D ,D ,−1 n 2 2 i even i odd In the following, we use the notation r to stand for a zigzag of length r on C ,ashiftof e e the points of C enclosed in the rectangle with opposite corners (1, −1) and (r, −2) and the notation r for a zigzag of length r on C , a shift of the points enclosed within the o o e e same rectangle on C . Given ω ∈ H , the map ψ stacks (ω − I{i even}) copies of the o −i zigzag i (whenever this is non-negative) vertically in increasing order and then removes a point from the bottom of each of the columns 1, 2, ... ,i, for each even i such that ω = 0 −i (giving the generalised Young diagram of an element of GFP (n)). Given ω ∈ H , D ,D ,0 2 2 the map ψ stacks (ω − I{i odd}) copies of the zigzag i (whenever this is non-negative) −i o vertically in increasing order and then removes a point from the bottom of each of the columns 1, 2, ... ,i, for each odd i such that ω = 0 (giving the generalised Young diagram −i of an element of GFP (n)). See Fig. 7 for examples of these maps. It is possible, but D ,D ,−1 2 2 tedious, to verify that these maps provide the necessary bijections. We leave this to the reader. Balázs et al. Res Math Sci (2022) 9:48 Page 21 of 46 48 ω =1 −7 n =19 ω =2 −5 m =4 ω =2 −4 −→ ω =1 −3 −11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 ω =(3, 0, 1, 2, 2, 0, 1, 1, 0, 1, 0, ...) ω =3 −1 ⏐ ⏐ e Since ω = 0, remove ⏐ −2 ⏐ 19,4 a point from the bottom of columns 1 and 2. n =19 m =4 Since ω = 0, remove −6 a point from the bottom of columns 1 through 6. ←− (a) A state ω ∈H and its image in GFP (19) D ,D ,0,4 2 2 ω =1 −8 n =19 ω =2 −6 m =4 ω =1 −4 −→ ω =2 −11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 −2 ω =(2, 2, 0, 1, 1, 2, 0, 1, 1, 0, 1, ...) ω =1 −1 ⏐ ⏐ Since ω = 0, remove −3 ⏐ ⏐ a point from the bottom 19,4 of columns 1,2 and 3. n =19 m =5 Since ω = 0, remove −7 a point from the bottom of columns 1 through 7. ←− − 3110 5 3110 (b) A state ω ∈H and its image in GFP (19) D ,D ,−1,5 2 2 e o Fig. 7 Bijections ψ and ψ 19,4 19,4 48 Page 22 of 46 Balázs et al. Res Math Sci (2022) 9:48 2,15 −→ 422 6441 0 0 220 −−−− 00 (a) An element of GFP (15) and its image in GFP (21) D ,D ,0 D ,D ,4 2 2 2 2 1,15 −→ − 422 644 0 0 2 200 −−− 00 (b) An element of GFP (15) and its image in GFP (19) D ,D ,−1 D ,D ,3 2 2 2 2 e o Fig. 8 Bijections φ and φ 2,15 1,15 We now consider the other values of the offset k. To tackle these cases it suffices, as in the Jacobi triple product case, to prove the following equivalences of generating functions: ⎨ (+1) f (˜q, t)˜q if k = 2, D ,D ,0 2 2 f (˜q, t) = D ,D ,k 2 2 2 ⎩ (+1) f (˜q, t)˜q if k = 2 + 1. D ,D ,−1 2 2 These follow naturally from a generalisation of Wright’s bijection φ , i.e. bijections φ :GFP (n) → GFP (n + ( + 1)), D ,D ,0 D ,D ,2 ,n 2 2 2 2 o 2 φ :GFP (n) → GFP (n + ( + 1) ) D ,D ,−1 D ,D ,2+1 ,n 2 2 2 2 defined for each n ≥ 0and  ∈ Z as follows. Given an element of GFP (n), the map D ,D ,0 2 2 |2|(|2|+1) φ adds the points inside the right angled triangle of size to either the left ,n 2 or top edge of its generalised Young diagram, depending on whether  ≥ 0or < 0, respectively. It then uses the new leading diagonal implied by the triangle to read off an element of GFP (n + ( + 1)). Given an element of GFP (n), the map φ D ,D ,2 D ,D ,−1 2 2 2 2 ,n |2+1|(|2+1|+1) adds the points inside the right-angled triangle of size to either the left or top edge of its generalised Young diagram, depending on whether  ≥ 0or < 0, respectively. It then uses the new leading diagonal implied by the triangle to read off an element of GFP (n + ( + 1) ). In both cases, the offset is supplied by the empty D ,D ,2+1 2 2 rows/columns coming from the triangle (in a similar fashion to Wright’s bijection). See e o Fig. 8 for examples of φ and φ .Notealsothatinbothcases thenumber m of distinct parts is clearly preserved and so the above maps restrict to give well defined bijections φ :GFP (n) → GFP (n + ( + 1)) D ,D ,0,m D ,D ,2,m ,n,m 2 2 2 2 o 2 φ :GFP (n) → GFP (n + ( + 1) ) D ,D ,−1,m D ,D ,2+1,m ,n,m 2 2 2 2 for each n, m ≥ 0and  ∈ Z. This proves the required equivalences of generating functions and hence completes the combinatorial justification of the identities in Theorem 1.1. Balázs et al. Res Math Sci (2022) 9:48 Page 23 of 46 48 4 Specialising to certain models In this section, we specialise the work of Sect. 3 to the ASEP(q, 1), 2-exclusion and 3- state models. For each of these models, the three variable identities in Theorem 1.1 will specialise to give two variable identities related to other known Jacobi style identities of combinatorial significance. 4.1 ASEP(q, 1) We will consider first the asymmetric simple exclusion process ASEP(q, 1) on ,as described by Redig et al. in [4]. This has asymmetry parameter 0 < q < 1and jump rates given by η −η −3 i i+1 p(η , η ) = q [η ] [2 − η ] and i i+1 i q i+1 q η −η +3 i i+1 q(η , η ) = q [2 − η ] [η ] . i i+1 i q i+1 q n −n q −q Here [n] := ,a q-deformation of n (we have that [n] → n as q → 1). q q −1 q−q The above rates might look strange at first sight, but they are carefully constructed in order to equip the corresponding process with “U (gl )-symmetry” (here U (gl )isthe q 2 q 2 quantum group associated with the Lie algebra gl ). The original motivation for construct- ing processes with “U (g)-symmetry” (for certain Lie algebras g) was the realisation that such processes have many explicit self-duality functions. The Markov generators for such processes are built by applying ground state transformations to quantum Hamiltonians associated with certain tensor products of representations of U (g). Roughly speaking the dimension of the representation relates to the number of particles allowed at a site, the dimension of g relates to the number of species of particle and tensors let us describe multi-site states. The Quantum Hamiltonian determines the jump rates. In the case of g = gl , the representations are the spin 2j representations for j ∈ Z (up to twist) and so onegetsafamilyASEP(q, j) of processes, each allowing up to 2j particles at a site. When j = , we recover classical ASEP and when j = 1 we get the process described above. We show that ASEP(q, 1) is a member of the blocking family. Conditions (B1) and (B2) clearly hold and the jump rates satisfy: (a) p(y, z) > q(z, y) for all y ∈{1, 2} and z ∈{0, 1}, p(1,0) p(2,1) −4 (b) = q = , q(0,1) q(1,2) 4 2 q [2] p(1,0)p(2,1)q(1,1)q(0,2) q −8 (c) = q · = 1. −4 2 q(0,1)q(1,2)p(2,0)p(1,1) q [2] Thus, (B3) is satisfied with the constants −2 −4 2 q [2] q q [2] 1 q q p = = and q = = asym asym −2 2 −4 −2 2 −4 [2] q + q 1 + q [2] q + q 1 + q q q and the functions y−z−2 −4 [z] q (1 + q )[3 − y] [3 − z] q q q f (z) = s(y, z) = . −4 [3 − z] q In this case, q˜ = q ,t = [2] and we have a one parameter family of product blocking measures of the form 0 η (2−η ) ∞ η (2−η ) i i −4η (i−c) i i 4(2−η )(i−c) i  i [2] q [2] q q q μ (η) = . −4(i−c) −8(i−c) 4(i−c) 8(i−c) (1 + [2] q + q ) (1 + [2] q + q ) q q i=−∞ i=1 Substituting the values of q˜ and t into Theorem 1.1 gives 4 4(+1) 2 4i 8i 2 2 S (q , [2] )q z = (1 + [2] q z + q z ) even q q ∈Z i≥1 48 Page 24 of 46 Balázs et al. Res Math Sci (2022) 9:48 4(i−1) −1 8(i−1) −2 ×(1 + [2] q z + q z ) 4i 8i 2 + (1 − [2] q z + q z ) i≥1 4(i−1) −1 8(i−1) −2 ×(1 − [2] q z + q z )) 4 4(+1) 2+1 4i 8i 2 2[2] S (q , [2] )q z = (1 + [2] q z + q z ) q odd q q i≥1 ∈Z 4(i−1) −1 8(i−1) −2 ×(1 + [2] q z + q z ) 4i 8i 2 − (1 − [2] q z + q z ) i≥1 4(i−1) −1 8(i−1) −2 ×(1 − [2] q z + q z )). −1 Since [2] = q +q , the quadratic terms on the RHS all factor and the products collapse as follows: 4i−1 4i+1 4i−5 −1 4i−3 −1 (1 + q z)(1 + q z)(1 + q z )(1 + q z ) i≥1 4i−1 4i+1 4i−5 −1 4i−3 −1 ± (1 − q z)(1 − q z)(1 − q z )(1 − q z ) i≥1 −1 −1 1 + q z 2i−1 2i−1 −1 = (1 + q z)(1 + q z ) 1 + qz i≥1 −1 −1 1 − q z 2i−1 2i−1 −1 ± (1 − q z)(1 − q z ) 1 − qz i≥1 ⎛ ⎞ 2i−1 2i−1 −1 2i−1 2i−1 −1 ⎝ ⎠ = (1 + q z)(1 + q z ) ∓ (1 − q z)(1 − q z ) . qz i≥1 i≥1 We have therefore proved the following identities. Theorem 4.1 4 (2+1) 2+1 2i−1 2i−1 −1 2 S (q , [2] )q z = (1 + q z)(1 + q z ) even q ∈Z i≥1 2i−1 2i−1 −1 − (1 − q z)(1 − q z ) i≥1 2 4 (2) 2 2i−1 2i−1 −1 2(1 + q ) S (q , [2] )q z = (1 + q z)(1 + q z ) odd q ∈Z i≥1 2i−1 2i−1 −1 + (1 − q z)(1 − q z ). i≥1 The RHS of these identities should look familiar. The first term is part of the product side of the Jacobi triple product, as seen in introduction (an alternative form is given in Sect. 3.3). Indeed, these two identities are isolating the odd/even terms, respectively (as mentioned in Sect. 3.3, this is the only reason for the sign changes on the RHS). Balázs et al. Res Math Sci (2022) 9:48 Page 25 of 46 48 We can now proceed in two ways. If we assume the Jacobi triple product, then Theorem 4.1 proves the following closed forms for the specialised normalising factors: 4 2 S (q , [2] ) =  (= f (q )), even q D ,D ,0 1 1 2j (1 − q ) j≥1 1 1 4 2 S (q , [2] ) =  = f (q ) . odd q D ,D ,0 1 1 2 2j 2 (1 + q ) (1 − q ) (1 + q ) j≥1 On the other hand, if we were able to find probabilistic explanations for these two equal- ities, then this would give a new probabilistic proof of the Jacobi triple product. However, the authors were unable to find such an explanation. To elaborate further, ASEP(q, 1) is a particular 0–1–2 system with blocking measure. However, the above equalities suggest that this blocking measure (and the one for its stood up process) should be intimately related to that of the 0-1 system ASEP(q, ) (i.e. classical ASEP) and its stood up process AZRP (which together prove the Jacobi triple product). Given the strange nature of the rates of ASEP(q, 1) it is not clear, at least to the authors, why this relationship should be expected. We can give explicit combinatorial explanations for these equalities. Let us look at the even case first. We have already seen that S (˜q, t) = f (˜q, t). Note that setting even D ,D ,0 2 2 4 −1 n m 4n −1 m 4n+m 4n+(m−2) q˜ = q and t = q + q sends q˜ t to q (q + q ) = q + q + ··· + 4n−(m−2) 4n−m q + q and so in order to prove the equality it suffices to show how m−1 elements of GFP (n) can be used to construct unique ordinary partitions of D ,D ,0,m 2 2 4n −(m −2i) into even parts for each 0 ≤ i ≤ m. Equivalently we must show how to obtain m m−2i unique ordinary partitions of 2n − . In order to do this, we take the generalised i 2 Young diagram corresponding to such a GFP, and for each square on the diagonal we colour in all intermediate white squares to the right and below. This almost creates a valid Young diagram but repeats in the rows of the GFP cause a problem, since the first entry in a repeat gives a row/column that is too short. To fix this, we add an extra black square to the end of such rows/columns, creating a valid Young diagram for a partition of 2n − . The other required partitions are obtained by adding more dots to this Young diagram. The only rows/columns that we are guaranteed to add dots to and still get a valid Young diagram are those corresponding to distinct entries in the rows of the GFP. For each of the choices of i such entries, we can add a dot onto the corresponding rows/columns, m−2i giving unique partitions of 2n − (for each 0 ≤ i ≤ m). See Fig. 9 for an example of the above construction (the colour grey is used to emphasise the squares that have been coloured black). The odd offset case is similar. We have already seen that tS (˜q, t) = f (˜q, t). Set- D ,D ,−1 odd 2 2 m m 4 −1 n m 4n+m 4n+(m−2) 4n−(m−2) ting q˜ = q and t = q +q sends q˜ t to q + q +···+ q + 1 m−1 4n−m 2 q as before. Multiplying by an extra q (to make the factor (1 + q ) on the LHS) gives 4n+(m+1) m 4n+(m−1) m 4n−(m−3) 4n−(m−1) q + q + ··· + q + q . In order to prove the 1 m−1 equality, it suffices to show how elements of GFP (n) can be used to construct D ,D ,0,−1 2 2 unique ordinary partitions of 4n − (m − 1 − 2i) into even parts for each 0 ≤ i ≤ m. Equiv- m m−1−2i alently we must show how to obtain unique ordinary partitions of 2n − .We i 2 do this in the same way as the even case, but being careful to also colour the intermediate squares in the first column and add a dot to the first column of the Young diagram if b is 1 48 Page 26 of 46 Balázs et al. Res Math Sci (2022) 9:48 −→ −→ (a) An element of GFP (14) and the associated partition of 2(14) − =27. D ,D ,0,2 2 2 {a } {b } {a ,b } 1 3 1 3 (b) Partitions of 28 and 29 coming from subsets of distinct entries of the GFP. Fig. 9 Partitions of 27, 28 and 29 coming from an element of GFP (14) D ,D ,0,2 2 2 −→ −→ − 422 3 200 3−1 (a) An element of GFP (16) and the associated partition of 2(16)− = 31. D ,D ,−1,3 2 2 {a } {b } {b } {a ,b } 1 1 2 1 1 {a ,b } {b ,b } {a ,b ,b } 1 2 1 2 1 1 2 Fig. 10 Partitions of 31, 32, 33 and 34 coming from an element of GFP (16) D ,D ,−1,3 2 2 not a distinct entry in row 2 of the GFP (the first column doesn’t correspond to a diagonal square). See Fig. 10 for an example. 4.2 Asymmetric particle-antiparticle exclusion process (3-state model) We now consider the asymmetric particle-antiparticle process with I ={−1, 0, 1},  = Z, asymmetry parameter 0 < q < 1, annihilation parameters a, a > 0, creation parameters γ , γ > 0 and jump rates as follows: • Particles (η = 1) jump left with rate q or right with rate 1 to empty sites, • Antiparticles (η =−1) jump left with rate 1 or right with rate q to empty sites, i Balázs et al. Res Math Sci (2022) 9:48 Page 27 of 46 48 Table 3 Jump rates p(η , η ) and q(η , η ), respectively, of i i+1 i i+1 the 3-state model η = 0 η = 1 η = 2 i+1 i+1 i+1 η =0 000 η =11 γ 0 η =2 210 η = 0 η = 1 η = 2 i+1 i+1 i+1 η =00 q 2q η =10 γqq η =2 000 • A particle–antiparticle neighbouring pair can be annihilated with rate a if η = 1and η =−1orrate a q if η =−1and η = 1, i+1 i i+1 • A particle-antiparticle pair can be created at two neighbouring empty sites with rate (i,i+1) (i,i+1) (i,i+1) (i,i+1) γ if η =−1and η = 1orrate γ q if η = 1and η =−1. i i+1 i i+1 Remark This process has been studied in the literature (see [2], for example), and it is known that the system can only have an i.i.d stationary distribution when the annihilation rate is twice the jump rate of a particle, i.e. when a = a = 2. We will consider this case from now on. We can view the 3-state model as a process on , with annihilation being a jump from a two particle site to an empty site and creation a jump from a one particle site to another one particle site. When thought of in this way, the jump rates are as given in Table 3. In order to be a member of the blocking family, we see that (B1) gives no constraints, (B2) is satisfied if and only if 0 <γ , γ ≤ 1, and the following must be satisfied: (a) p(y, z) > q(z, y) for all y ∈{1, 2} and z ∈{0, 1}, p(1,0) p(2,1) −1 (b) = q = , q(0,1) q(1,2) p(1,0)p(2,1)q(1,1)q(0,2) γ (c) 1 = = , i.e. 0 <γ = γ ≤ 1. q(0,1)q(1,2)p(2,0)p(1,1) Under this assumption, (B3) is then satisfied with the constants 1 q p = and q = asym asym 1 + q 1 + q and the functions (1 + q)if y = z = 1, ⎪ γ ⎧ ⎪ ⎪ ⎪ 0if z = 0, 2γ (1 + q)if y = 1and z = 2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨  ⎨ if z = 1, f (z):= s(y, z):= 2γ (1 + q)if y = 2and z = 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 ⎪ γ if z = 2, ⎪ (1 + q)if y = z = 2, γ ⎪ 2 0if y = 3or z = 3. 48 Page 28 of 46 Balázs et al. Res Math Sci (2022) 9:48 In this case, q˜ = q and t = and we have a one-parameter family of blocking measures of the form 1   1 I{η =1} I{η =1} i i 2 2 2 −(i−c)η 2 (2−η )(i−c) 0 i ∞ i q q γ γ μ (η) =   . 2 2 −(i−c) −2(i−c) (i−c) 2(i−c) 1 + q + q 1 + q + q i=−∞ i=1 γ γ Since γ can be almost freely chosen (there is only the constraint that 0 <γ ≤ 1), applying Theorem 1.1 to this subfamily will not tell us anything new (we will get the same three variable identity but with domain restricted to t ≥ 2). However, certain members of this subfamily can give interesting two variable identities. As an example, we will choose the creation parameter to be γ = (the inverse of the annihilation parameter a = 2). For this process, we then have q˜ = q and t = 2. Substituting these values of q˜ and t into Theorem 1.1 gives (+1) 2 i 2i 2 i−1 −1 2(i−1) −2 2 S (q, 2)q z = (1 + 2q z + q z )(1 + 2q z + q z ) even ∈Z i≥1 i 2i 2 i−1 −1 2(i−1) −2 + (1 − 2q z + q z )(1 − 2q z + q z )), i≥1 (+1) 2+1 i 2i 2 i−1 −1 2(i−1) −2 4 S (q, 2)q z = (1 + 2q z + q z )(1 + 2q z + q z ) odd ∈Z i≥1 i 2i 2 i−1 −1 2(i−1) −2 − (1 − 2q z + q z )(1 − 2q z + q z )). i≥1 Once again the quadratic terms on the RHS all factor and so we have proved the following identities Theorem 4.2 (+1) 2 i i−1 −1 2 2 S (q, 2)q z = ((1 + q z)(1 + q z )) even i≥1 ∈Z i i−1 −1 2 + ((1 − q z)(1 − q z )) i≥1 (+1) 2+1 i i−1 −1 2 4 S (q, 2)q z = ((1 + q z)(1 + q z )) odd ∈Z i≥1 i i−1 −1 2 − ((1 − q z)(1 − q z )) . i≥1 Again the first term on the RHS looks familiar and is part of the square of the product side of the Jacobi triple product. Indeed, if we assume Jacobi triple product, then we find that 1 k(k+1) i i−1 −1 2 k ((1 + q z)(1 + q z )) =  q z (1 − q ) i≥1 i≥1 k∈Z k (k +1) (k−k )(k−k +1) + k 2 2 = q z . i 2 (1 − q ) i≥1 k∈Z k ∈Z Balázs et al. Res Math Sci (2022) 9:48 Page 29 of 46 48 The term corresponding to k = 2 can be written as (using Jacobi triple product in the second equality, written as in the introduction but setting z = 1) k (k +1) (2−k )(2−k +1) + 2 2 2 q z i 2 (1 − q ) i≥1 k ∈Z 1  2 (k −) (+1) 2 =  q q z i 2 (1 − q ) i≥1 k ∈Z ⎛ ⎞ 2i−1 2 2i (1 + q ) (1 − q ) (+1) 2 ⎝ ⎠ = q z i 2 (1 − q ) i≥1 ⎛ ⎞ 2i−1 2 i (1 + q ) (1 + q ) (+1) 2 ⎝ ⎠ = q z . (1 − q ) i≥1 Similarly the term corresponding to k = 2 + 1 can be written as (using Jacobi triple product in the second equality, written as in Sect. 3.3 but setting z = 1) k (k +1) (2−k +1)(2−k +2) + 2+1 2 2 q z i 2 (1 − q ) i≥1 k ∈Z 1   2 (k −)(k −−1) (+1) 2+1 =  q q z i 2 (1 − q ) i≥1 k ∈Z ⎛ ⎞ 2i 2 2i (1 + q ) (1 − q ) 2 (+1) 2+1 ⎝ ⎠ = 2 q z i 2 (1 − q ) i≥1 ⎛ ⎞ 2i 2 i (1 + q ) (1 + q ) 2 (+1) 2+1 ⎝ ⎠ = 2 q z . (1 − q ) i≥1 Thus, we see that, assuming Jacobi triple product, the identities are equivalent to the equalities 2i−1 2 i (1 + q ) (1 + q ) S (q, 2) = , even (1 − q ) i≥1 2i 2 i (1 + q ) (1 + q ) S (q, 2) = . odd (1 − q ) i≥1 As in the previous section, it would be interesting to find purely probabilistic proofs of these product forms. The more complicated nature of the products suggests that this is non-trivial. Alternatively one could have not assumed Jacobi triple product in the above, but instead assumed Theorem 4.1. The identities would then give (non-trivial) relations between the 4 4 four specialised normalising factors S (q, 2),S (q, 2),S (q , [2] )and S (q , [2] ) even odd even q odd q that we would have proved probabilistically without additional assumptions. While the identities in Theorem 4.2 might seem unnatural, they do have a combinatorial i 2 interpretation in terms of 2-coloured GFP’s. The product (1 + q ) is the generating i≥1 function for coloured partitions of n into red/blue parts where each red/blue part appears 48 Page 30 of 46 Balázs et al. Res Math Sci (2022) 9:48 at most once (for example we allow 5 = 2 + 2 + 1but not5 = 2 + 2 + 1). The pair of 2-coloured partitions 2 +2 +1and 2 +2 +1 are counted as the same in the above product, and so to avoid overcounting we favour a particular colour when listing repeats. By the “General Principle” of Andrews, we have that: i i−1 −1 2 k ((1 + q z)(1 + q z )) = f (q)z , C ,C ,k 2 2 i≥1 k∈Z where f (q) is the generating function for the sets GFP (n)ofGFP’s of n with C ,C ,k C ,C ,k 2 2 2 2 offset k and each row being a 2-coloured partition. The content of the above identities is then that (+1) S (q, 2)q if k = 2, even f (q) = C ,C ,k 2 2 2 (+1) 2S (q, 2)q if k = 2 + 1. odd Thus, the two specialised normalising factors satisfy f (q) = 2S (q, 2) and C ,C ,−1 2 2 odd f (q) = S (q, 2) and so have a natural combinatorial interpretation. These two base C ,C ,0 even 2 2 cases are both explicitly clear since we know from Sect. 3.3 that S (˜q, t) = f (˜q, t) even D ,D ,0 2 2 and tS (˜q, t) = f (˜q, t), and each element of GFP (n)and GFP (n) D ,D ,−1 D ,D ,0,m D ,D ,−1,m odd 2 2 2 2 2 2 can be 2-coloured in 2 ways (exactly what is counted when setting t = 2). The other offset cases can be proved in the usual fashion, by the equalities of generating functions: ⎨ (+1) f (q)q if k = 2, C ,C ,0 2 2 f (q) = C ,C ,k 2 2 (+1) f (q)q if k = 2 + 1. C ,C ,−1 2 2 e o (The proof follows from the maps φ , φ of Sect. 3.3, but by using coloured generalised ,n ,n Young diagrams). The function f (q) is the function C (q) defined by Andrews on p.7 of [1]. Indeed, C ,C ,0 2 2 2 the specialised normalising factor is: 2 3 4 5 6 7 8 S (q, 2) =1+4q +9q +20q + 42q + 80q + 147q + 260q + 445q + ··· even which agrees with the expansion of C (q) found on p.8 of the same book. In Corollary 5.2 of this book, Andrews uses Jacobi triple product to prove a product formula for C (q), which is equivalent to the product formula we found above for S (q, 2). Andrews’ book even only considers GFP’s with offset 0 and so does not give a similar analysis of the function f (q). However, using MAGMA we were able to compute that C ,C ,−1 2 2 2 3 4 5 6 7 8 f (q) =2+4q +12q + 24q + 50q + 92q + 172q + 296q + 510q + ··· C ,C ,−1 2 2 which agrees with 2S (q, 2), as expected, and appears to have the same coefficients as odd twice OEIS sequence A137829 [7], implying the product form we derived above. The function C (q)isone of afamilyoffunctions C (q), counting GFP’s of offset 0 with rows having a similar condition to the above but with k colours. The whole family of functions is studied in Andrews’ book. In general they are not given by products but can be shown to be sums of products. It would be interesting to know whether there exists a k- state model for each k, whose stood up process and stationary blocking measures provide normalising factors relating to these functions (and their corresponding non-zero offset analogues). 4.3 Asymmetric 2-exclusion Now we consider the asymmetric 2-exclusion process on  with asymmetry parameter 0 < q < 1. The nonzero left/right jump rates are Balázs et al. Res Math Sci (2022) 9:48 Page 31 of 46 48 p(η , η ) = I{η = 0}I{η = 2} and q(η , η ) = qI{η = 0}I{η = 2}. i i+1 i i+1 i i+1 i+1 i We show that this is a member of the blocking family. Conditions (B1) and (B2) clearly hold and the jump rates satisfy: (a) p(y, z) > q(z, y) for all y ∈{1, 2} and z ∈{0, 1}, p(1,0) p(2,1) −1 (b) = q = , q(0,1) q(1,2) p(1,0)p(2,1)q(1,1)q(0,2) (c) = 1. q(0,1)q(1,2)p(2,0)p(1,1) Thus, (B3) is satisfied with the constants 1 q p = and q = , asym asym 1 + q 1 + q and the functions, (1 + q) for y, z ∈{1, 2}, f (z) = I{z = 0},s(y, z) = 0if y = 3or z = 3. In this case, q˜ = q, t = 1 and we have a one-parameter family of product stationary blocking measures of the form −η (i−c) (2−η )(i−c) i  i q q μ (η) = . −(i−c) −2(i−c) (i−c) 2(i−c) (1 + q + q ) (1 + q + q ) i≤0 i≥1 Substituting the values of q˜ and t into Theorem 1.1 gives Theorem 4.3 (+1) 2 i 2i 2 i−1 −1 2(i−1) −2 2 S (q, 1)q z = 1 + q z + q z 1 + q z + q z even ∈Z i≥1 i 2i 2 i−1 −1 2(i−1) −2 + 1 − q z + q z 1 − q z + q z i≥1 (+1) 2+1 i 2i 2 i−1 −1 2(i−1) −2 2 S (q, 1)q z = 1 + q z + q z 1 + q z + q z odd ∈Z i=1 i 2i 2 i−1 −1 2(i−1) −2 − 1 − q z + q z 1 − q z + q z . i=1 It is clear that this specialisation has a natural combinatorial meaning. Recall that in Sect. 3.3 we used the “General Principle” to expand i 2 2i −1 i−1 −2 2(i−1) k (1 + tzq˜ + z q˜ )(1 + tz q˜ + z q˜ ) = f (˜q, t)z D ,D ,k 2 2 i≥1 k∈Z with f (˜q, t) being the two variable generating function for the sets GFP (n) D ,D ,k D ,D ,k,m 2 2 2 2 defined earlier. Setting t = 1 is naturally interpreted as not distinguishing GFP’s by their number of distinct parts per row, i.e. f (˜q, 1) = f (˜q), the one variable generating D ,D ,k D ,D ,k 2 2 2 2 function for the sets GFP (n). The content of the above identities is then that D ,D ,k 2 2 ⎨ (+1) S (q, 1)q if k = 2, even f (q) = D ,D ,k 2 2 2 (+1) S (q, 1)q if k = 2 + 1. odd 48 Page 32 of 46 Balázs et al. Res Math Sci (2022) 9:48 e o e o The maps ψ , ψ , φ and φ of Sect. 3.3 give explicit proofs for all of these equalities, as n n ,n ,n expected. Let’s consider the two base cases in more detail. The function f (q) is the function D ,D ,0 2 2 (q) defined by Andrews on p.6 of [1]. The specialised even normalising factor is: 2 3 4 5 6 7 8 S (q, 1) = 1 + q + 3q + 5q + 9q + 14q + 24q + 35q + 55q + ··· even which agrees with the expansion of  (q) on p.7 of the same book, as expected. An interesting result in this book is Corollary 5.1, which uses the Jacobi triple product and other results to prove that (q) =  . i 12i−10 12i−9 12i−3 12i−2 (1 − q )(1 − q )(1 − q )(1 − q )(1 − q ) i≥1 So the specialised even normalising factor can be expressed as the above product. Andrews’ book only considers GFP’s with offset 0 and so does not give a similar analysis of the function f (q). However, using MAGMA we were able to compute that D ,D ,−1 2 2 2 3 4 5 6 7 8 f (q) = 1 + 2q + 3q + 6q + 10q + 16q + 26q + 40q + 60q + ··· D ,D ,−1 2 2 which agrees with S (q, 1) and appears to have the same coefficients as OEIS sequence odd A201077 [8], implying that S (q, 1) = f (q) odd D ,D ,−1 2 2 = . 2i−1 2 12i−8 12i−6 12i−4 12i (1−q ) (1−q )(1−q )(1−q )(1−q ) i≥1 This discussion raises two interesting questions. Firstly, is there a probabilistic expla- nation for the above product forms? These products are quite complicated and it is not clear a priori that we should expect such a factorisation. It could be possible that there is an alternative way to stand up the 2-exclusion process, giving a more natural normalising factor. Secondly, in this case, the ASEP(q, 1) case and the 3-state model case the specialised normalising factors appear to be products. Could it be that the unspecialised normalising factors S (˜q, t)and S (˜q, t) are products? even odd 5 Asymmetric k-exclusion It is natural to ask whether we can generalise the results of this paper to higher order particle systems with blocking measure. Unfortunately, as remarked in Sect. 3.2 the “stood up” process in general is not guaranteed to have a product stationary blocking measure. However, we will now see that the asymmetric k-exclusion processes for k ≥ 1are sufficiently well behaved and lead to an interesting family of combinatorial identities, generalising the ones found in the 2-exclusion section. The asymmetric k-exclusion process is the particle system with I ={0, 1, ... ,k},  = Z, asymmetry parameter 0 < q < 1 and jump rates p(η , η ) = I{η = 0}I{η = k} and q(η , η ) = qI{η = 0}I{η = k}. i i+1 i i+1 i i+1 i+1 i We now show that this is member of the blocking family. Conditions (B1) and (B2) clearly hold so it suffices to check (B3). We see that the jump rates are described by the constants 1 q p = q = asym asym 1 + q 1 + q and functions (1 + q) for y, z ∈{1, 2, ... ,k}, f (z) = I{z = 0} and s(y, z) = 0if y = k + 1or z = k + 1. Balázs et al. Res Math Sci (2022) 9:48 Page 33 of 46 48 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 0 −1 (a) Ground state η (b) Ground state η −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −2 −3 (c) Ground state η (d) Ground state η Fig. 11 Four ground states of 4-exclusion Thus, k-exclusion is a member of the blocking family and by Theorem 2.1 has one param- eter family of product stationary blocking measures 0 ∞ −(i−c)η (k−η )(i−c) i  i q q μ (η) = , c c k(i−c) Z (q) q Z (q) i i i=−∞ i=1 c −(i−c)y where Z (q) = q is the normalising factor. y=0 By definition of the state space, every η ∈  hasaleftmostparticleand aright most hole, i.e. η = 0 for small enough i and η = k for big enough i. By asymmetry, it then i i −m follows that the ground states of  are all shifts of the following k ground states η , for m ∈{0, 1, ... ,k − 1} with 0if i < 0, −m η = m if i = 0, k if i > 0. See Fig. 11 for example when k = 4. 5.1 Ergodic decomposition of ∞ 0 The quantity N(η):= (k − η ) − η is finite and is conserved by the dynamics i i i=1 i=−∞ of the system. Thus, just as in Sect. 3.1 we can decompose  =  , into irreducible n∈Z components  :={η ∈  : N(η) = n}. Note now that the left shift operator τ gives a n n−k n bijection  − →  (i.e. if η ∈  then, N(τη) = n − k). −m −m Remark Since N(η ) =−m for each m ∈{0, 1, ... ,k − 1},the shifts of η have con- served quantity in kZ−m and give the ground states for the (−m mod k)part n∈kZ−m of . n,c c n We now calculate ν (·):= μ (·|N(·) = n), the unique stationary distribution on  . Lemma 5.1 The following relation holds c kc−N(η) c μ (τη) = q μ (η). This gives the recursion c n−kc c μ ({N = n}) = q μ ({N = n − k}). 48 Page 34 of 46 Balázs et al. Res Math Sci (2022) 9:48 The proof is similar to that of Lemma 3.1 (both claims are special cases of Lemma 6.1 and Corollary 6.2 in [3]). The general solution of this recursion is (n+m)(n+k−m) c −(n+m)c c 2k μ ({N = n}) = q μ ({N =−m}) if n ∈ kZ − m with m ∈{0, 1, ... ,k − 1}. Since there is a dependence on the class of n modulo k we will need to calculate the probabilities μ ({N(η) ≡−m mod k}) for m ∈{0, 1, ... ,k − 1} in order to finish our n,c calculation of ν . Lemma 5.2 ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ k−1 ∞ k−1 rj c −rm c ⎝ ⎝ ⎝ ⎠ ⎠ ⎠ μ ({N ≡−m mod k}) = 1 + ζ 1 + (ζ − 1)μ (j) , k k r=1 i=−∞ j=1 th where ζ is a primitive k root of unity. Proof Define the partial conserved quantity a 0 N (η) = (k − η ) − η for a ≥ 1 a i i i=1 i=−a and note that N (η) → N(η)as a →∞. Consider the character group Z/kZ := Hom Z/kZ, C ={χ , χ , ... , χ }, 0 1 k−1 where χ (1) = ζ for i ∈{0, 1, ... ,k − 1} and ζ aprimitive k-th root of unity. Note that i k χ = χ . For each a ≥ 1 we define the random variables a a Y := χ (−N (η)) = χ η = χ (η ). a 1 a 1 i 1 i i=−a i=−a Since η → 0as i →−∞ and η → k as i →∞ we have that Y → Y as a →∞, where i i a ∞ ∞ Y := χ (−N(η)) = χ ( η ) = χ (η ). 1 1 i 1 i i=−∞ i=−∞ c r We now compute the moments E [Y ] for r ∈{0, 1, ... ,k − 1} (the expectation being with respect to μ ). Since |Y |= 1 for all a ≥ 1, dominated convergence applies and by the product structure of μ we have that ∞ k ∞ k c r c r r c c E [Y ] = lim E [Y ] = χ (j) μ (j) = χ (j)μ (j) . 1 r a i i a→∞ i=−∞ j=0 i=−∞ j=0 On the other hand, we have that k−1 c r r c E [Y ] = χ (m) μ − N(η) ≡ m mod k m=0 k−1 = χ (m)μ N(η) ≡−m mod k . m=0 Balázs et al. Res Math Sci (2022) 9:48 Page 35 of 46 48 Hence, we have the linear system of equations ∞ k k−1 c c χ (j)μ (j) = χ (m)μ N(η) ≡−m mod k . r r m=0 i=−∞ j=0 By orthogonality of characters, we get k−1 ∞ k c c μ (N(η) ≡−m mod k) = χ (m) χ (j)μ (j) r r r=0 i=−∞ j=0 k−1 ∞ k = χ (−m) χ (j)μ (j) r r r=0 i=−∞ j=0 ⎛ ⎛ ⎞ ⎞ k−1 ∞ k rj −rm c ⎝ ⎝ ⎠ ⎠ = ζ ζ μ (j) k k r=0 i=−∞ j=0 ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ k−1 ∞ k−1 rj −rm ⎝ ⎝ ⎝ ⎠ ⎠ ⎠ = 1 + ζ 1 + (ζ − 1)μ (j) . k k i r=1 i=−∞ j=1 Remark A priori this appears to be complex-valued. However, it is in fact real valued since it is fixed by complex conjugation. When k = 2 this is clearly real valued and agrees with Lemma 3.2 (recalling that t = 1 for 2-exclusion). If we combine Lemma 5.1 and Lemma 5.2, we find the form of the unique stationary n,c n distribution ν on  . Proposition 5.3 For n ∈ kZ − mwithm ∈{0, 1, ... ,k − 1}, the stationary distribution on  is given by 0 ∞ k(+1)  −(i−c)η  (k−η )(i−c) q i q i −m−kc k q I{N(η) = n} k(i−c) Z (q) q Z (q) ∈Z i=−∞ i=1 n,c ν (η) =    . (n+m)(n+k−m) rj −(n+m)c k−1 −rm ∞ k−1 c 2k q 1 + ζ 1 + (ζ − 1)μ (j) r=1 i=−∞ j=1 i k k Remark As in Sect. 3.1, these distributions are independent of c but we will need to stress the dependence of both the numerator and denominator on c. k rj −j(i−c) ζ q rj k−1 j=0 c k Notice that 1 + (ζ − 1)μ (j) = for each r ∈{1, ... ,k − 1}. Defining j=1 k i Z (q) rj c −j(i−c) n,c W (q):= ζ q for each such r we can write ν as i,r j=0 n,c ν (η) 0 ∞ k(+1) −m−kc −(i−c)η (k−η )(i−c) i i k q q q I{N(η) = n} ∈Z i=−∞ i=1 =    . (n+m)(n+k−m) −(n+m)c k−1 −rm k(i−c) c c k(i−c) c c 2k q q Z (q)Z (q) + ζ q W (q)W (q) i≥1 −i+1 i r=1 i≥1 −i+1,r i,r when n ∈ kZ − m with m ∈{0, 1, ... ,k − 1}. c c Remark When k = 2, there is only the possibility r = 1and W (q) = W (q, 1), as i,1 i expected (here W (q, t) is the function defined in Sect. 3.1). The corresponding stationary distributions then agree with the ones for 2-exclusion given by Proposition 3.3 (after setting t = 1). 48 Page 36 of 46 Balázs et al. Res Math Sci (2022) 9:48 10 14 3 9 13 2 6 8 12 1 4 5 7 11 LMP RMH -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 (A) A 4-exclusion state η. (B) Its stood up state ω. Fig. 12 Standing up map on 4-exclusion −11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 −11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 (a) ω (b) ω −11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 −11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 −2 −3 (c) ω (d) ω Fig. 13 Four ground states of stood up 4-exclusion 5.2 Standing up/laying down In this section, we transfer the dynamics on  to that of a restricted particle system on <0 Z by using the same “standing up” method as described in Sect. 3.2. By doing this we ≥0 obtain an alternative characterisation of the stationary distributions given in Proposition 5.3. n th We recall the standing up map. Given η ∈  let S being the site of the r particle when reading left to right, bottom to top. The corresponding stood up state is then n <0 T (η) = ω ∈ Z ,with ω = S − S .See Fig. 12 for an example with k = 4. −r r+1 r ≥0 n <0 A priori the “standing up” map T is an injection into Z . However, since η ≤ k for ≥0 all i, the image of T lies in the restricted state space <0 H :={ω ∈ Z : ω = ω = ··· = ω = 0 ⇒ ω = 0, ∀i > 0}. −i −i−1 −i−(k−2) −i−(k−1) ≥0 Since η = k for i large ω must coincide far to the left with one of the following states −m −m ω , for m ∈{0, 1, ... ,k − 1},definedby ω = I{i ∈ kZ + m} (Fig. 13). More precisely, −i m is uniquely determined by n ∈ kZ − m. −m −m −m Remark Note that all shifts of the ground state η ∈  stand up to give ω for each m ∈{0, 1, ... ,k − 1}. This is in direct analogy with the 2-exclusion case. k−1 n −m −m We now see that the image of T lies in H := H , where the disjoint sets H are m=0 defined as −m −m H :={ω ∈ H : ∃N >0s.t ω = ω ∀i ≥ N }. −i −i −m Given ω ∈ H we let D (ω) be the minimum such N in the above. −m n n −m Lemma 5.4 For n ∈ kZ − mwithm ∈{0, 1, ... ,k − 1},wehaveT ( ) = H . Proof It suffices to show surjectivity of T for each n. Balázs et al. Res Math Sci (2022) 9:48 Page 37 of 46 48 Table 4 Jump rates p (ω , ω ) and q (ω , ω ), ω −r −r+1 ω −r −r+1 respectively ω ≥ 0 −r+1 ω =00 −r k−1 ω =11 − I{ω = 0} −r −r−j j=1 ω ≥21 −r ω = 0 ω = 1 ω ≥ 2 −r+1 −r+1 −r+1 k−1 ω ≥00 q(1 − I{ω = 0}) q −r −r+1+j j=1 Table 5 Boundary jump rates for the stood up process, ω Rate into the boundary Rate out of the boundary ω =00 q −1 k−1 ω =11 − I{ω = 0} −1 −r−j j=1 ω ≥21 −1 −m n Take n ∈ kZ − m and ω ∈ H , then construct the state η ∈  having LMP at site D (ω) −m n + D (ω) − I{D (ω) ∈ / kZ − m} −m −m S = + 1 − ω 1 −i i=1 th n n and r particle at site S = S + ω .Itisclear that T (η) = ω and hence T is r r−1 1−r surjective. Just as before we call these inverse maps the “laying down“ maps. Using the “standing up” maps we define a particle system on H whose dynamics are inherited from those on . In particular right jumps in η correspond to right jumps in ω and similarly for left jumps. The explicit right/left jump rates are given in Table 4 for r ≥ 2. Since the “stood up ” process is only defined on the negative half integer line, we must consider what happens at the boundary site. As in Sect. 3.2, we will consider an open infinite type boundary (Table 5). To find the stationary distribution for the “stood up” process, we first consider the ∗ <0 unrestricted process ω ∈ Z , i.e. the process described by the same jump rates as ω but ≥0 where the number of consecutive zeros is not restricted. It is clear that the unrestricted process is simply the zero-range process which is a member of the blocking family [3], with one parameter family of blocking measures given by the marginals ∗,cˆ (i+cˆ)z (i+cˆ) π (z) = q (1 − q ). −i By considering dynamics at the boundary, we can fix the value of cˆ and so have a sin- ∗,cˆ gle product blocking measure. We suppose that π satisfies detailed balance over this boundary edge, i.e. ∗,cˆ ∗,cˆ π (y) · “rate into the boundary” = π (y − 1) · “ rate out of the boundary” for all y ≥ 1. −1 −1 Thus, for all y ≥ 1wehave (1+cˆ)y (i−c) (1+cˆ)(y−1) (i−c) q (1 − q ) = q (1 − q )q 48 Page 38 of 46 Balázs et al. Res Math Sci (2022) 9:48 and hence cˆ = 0, giving the stationary blocking measure ∗ ∗ iω i −i π (ω ) = q (1 − q ). i≥1 Now that we have the stationary distribution for the unrestricted process we consider k−1 −m the restriction to H and find the stationary measure. Recall that H = H ,and m=0 −m −m note that for each H is the irreducible component of the ground state ω .Wedefine −m ∗ stationary measures π on these irreducible components in terms of π . It seems natural −m ∗ −m ∗ to define these measures as π (·) = π (·|· ∈ H ). However, w.r.t π the probability of being in any irreducible component is zero and so these quantities are undefined. To rectify this, we use a similar formal reasoning as we did in Sect. 3.2 in order to give the stationary distributions in the following form: −m −m φ (ω )I{ω ∈ H } −i −i i≥1 −m π (ω) = , −m φ (ω ) −i −i −m i≥1 ω ∈H where ∗ ⎨ i(ω −1) −i π (ω ) q if i ∈ kZ + m, −i −m −i φ (ω ) = = −i −i −m iω π (ω ) −i q otherwise. −i −i These are given explicitly in the following proposition. −m Proposition 5.5 For each m ∈{0, 1, ... ,k − 1} the unique stationary measure on H is iω + i(ω −1) −i −i i∈ /kZ+m i∈kZ+m −m π (ω) = , (k) S (q) −m iω + i(ω −1) −i −i (k) i∈ /kZ+m i∈kZ+m where S (q) = q is the normalising factor with respect to −m −m ω ∈H −m H . −m Proof The result once again follows from Proposition 5.10 of [6] since each π is a restriction of π . 5.3 Identities n n By Lemma 5.4, the standing up transformation T describes a bijection between −m n and H , for the unique value m ∈{0, 1, ... ,k − 1} such that n ∈ kZ − m.Since T preserves the dynamics of the corresponding processes, we get an equality of measures, n,c −m n ν (η) = π (T (η)) for this value of m. −m −m Recall that k-exclusion has k ground states up to shift; η ∈  for m ∈{0, 1, ... ,k − −m −m −m −m,c −m −m −m 1}, satisfying T (η ) = ω .Thus, ν (η ) = π (ω ) and so by Proposition 5.3 and Proposition 5.5 we have the following identities for m ∈{0, 1, ... ,k − 1} (after rearrangement): k(+1) (k) −m−(k−m)c k(i−c) c c k S (q)q = q Z (q)Z (q) −m −i+1 i ∈Z i≥1 Balázs et al. Res Math Sci (2022) 9:48 Page 39 of 46 48 ⎛ ⎞ k−1 −rm k(i−c) c c ⎝ ⎠ + ζ q W (q)W (q) . −i+1,r i,r r=1 i≥1 c c −c Writing Z (q)and W (q) explicitly and letting z = q proves the following identities. i i,r Theorem 1.3 For 0 < q < 1,z = 0 and m ∈{0, 1, ... ,k − 1} k(+1) (k) −m k−m k S (q)q z −m ∈Z ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ k−1 k k −rm −αr αi α αr α(i−1) −α ⎝ ⎝ ⎠ ⎝ ⎠ ⎠ = ζ ζ q z ζ q z . k k k r=0 i≥1 α=0 α=0 Remark When k = 2, these agree with the two identities coming from 2-exclusion in Theorem 4.3. We now discuss the combinatorial significance of these identities. Note first that i 2i k (1 + q + q + ··· q ) is the generating function for ordinary partitions of n with i≥1 each part appearing at most k times. By the “General Principle” of Andrews, we can then interpret the first term on the RHS as: i 2i 2 ki k i−1 −1 2(i−1) −2 k(i−1) −k (1 + q z + q z + ··· + q z )(1 + q z + q z + ··· + q z ) i≥1 = f (q)z D ,D ,k k k k ∈Z where f (q) is the generating function for GFP (n). The function f (q)is D ,D ,k D ,D ,k D ,D ,0 k k k k k k the function  (q)definedbyAndrews on p.6of[1] (the special case k = 2 appeared earlier when considering 2-exclusion). The case of nonzero offset is not considered in the book. The other terms on the RHS might look unwieldy at first glance but are in fact the above −r but with change of variables z → ζ z. We saw this in all previous examples; ζ =−1 and this was providing the sign changes in the identities, the purpose of which were to isolate odd/even terms of another identity. Here we have the same behaviour, but now we are using kth roots of unity to isolate the z terms where k lies in a fixed class mod k.To justify this we expand the whole RHS; for m ∈{0, 1, ... ,k − 1},wehave ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ k−1 k k −rm −αr αi α αr α(i−1) −α ⎝ ⎝ ⎠ ⎝ ⎠ ⎠ ζ ζ q z ζ q z k k r=0 i≥1 α=0 α=0 k−1 −rm −rk k = ζ ζ f (q)z D ,D ,k k k k k r=0 k ∈Z ⎛ ⎞ k−1 −r(m+k ) ⎝ ⎠ = ζ f (q)z D ,D ,k k k k k ∈Z r=0 = k f (q)z . D ,D ,k k k k ≡−m mod k Given the above, we now see that our identities are equivalent to the equalities k(+1) (k) −m f (q) = S (q)q if k = k − m with m ∈{0, 1, ... k − 1}. D ,D ,k −m k k 48 Page 40 of 46 Balázs et al. Res Math Sci (2022) 9:48 ω−8 =1 n =29 ω =2 −7 −→ −12−11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 ω =1 −5 ω =(3, 2, 0, 1, 1, 0, 2, 1, 1, 0, 0, 1, ...) ω =1 −4 ω =2 −2 (0) ψ ⏐ ω−1 =3 ⏐ Since ω =0, remove a −3 point from the bottom of columns 1,2 and 3. ←− Since ω = 0,removea −6 n =29 point from the bottom of columns 1 through 6. (a) A state ω ∈H and its image in GFP (29) D ,D ,0 3 3 ω =1 −8 n =25 ω =1 −6 −→ ω =1 −5 −13−12−11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 ω−4 =2 ω =(0, 2, 2, 2, 1, 1, 0, 1, 0, 1, 0, 0, 1...) ω =2 −3 ω =2 −2 (1) ψ ⏐ ⏐ ⏐ Since ω =0, remove −1 a point from the bottom of column 1. n =25 ←− Since ω =0, remove a −7 point from the bottom of columns 1 through 7. − 2100 66 4 2 0 −1 (b) A state ω ∈H and its image in GFP (25) D3,D3,−1 (0) (1) (2) Fig. 14 Bijections ψ , ψ and ψ (cont.) 29 25 19 Balázs et al. Res Math Sci (2022) 9:48 Page 41 of 46 48 ω =1 −7 n =19 ω =1 −6 −→ ω =1 −4 −12−11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 ω =(3, 0, 2, 1, 0, 1, 1, 1, 0, 0, 1, 0, ...) ω =2 −3 ω =3 ⏐ −1 (2) ψ ⏐ ⏐ Since ω = 0, remove a −2 point from the bottom of columns 1 and 2. n =19 ←− Since ω = 0, remove a −5 point from the bottom of columns 1 through 5. ω =0 −5 −− 1100 6 3 3100 −2 (c) A state ω ∈H and its image in GFP (19) D ,D ,−2 3 3 Fig. 14 continued Once again, we have proved this probabilistically but are also interested in explicit combi- natorial explanations. In a similar vein to 2-exclusion, we find that there are k base cases, (k) f (q) = S (q) for m ∈{0, 1, ... ,k −1}. These can be proved by adapting the maps D ,D ,−m −m k k e o ψ , ψ of Sect. 3.3 to give maps: n n ⎧ ⎫ ⎨ ⎬ (m) −m ψ : ω ∈ H : iω + i(ω − 1) = n → GFP (n), −i −i D ,D ,−m n k k ⎩ ⎭ i∈ /kZ+m i∈kZ+m for n ≥ 0and m ∈{0, 1, ... ,k − 1}. We must first describe the process of assigning generalised Young diagrams to GFP’s with the k-repetition condition. An element of GFP (n)with m ∈{0, 1, ... ,k − 1} D ,D ,−m k k can be assigned a generalised Young diagram on the set of points C ={(n ,n ) ∈ m 1 2 Z | n + n ≡ m mod k} as follows. The subset corresponding to such an element with 1 2 s entries on the top row consists of the s leading diagonal points (m + 1, −1), (m + 1 1 2, −2), ... , (m + s , −s ), the first a points of C to the right of (m + i, −i) and the first b 1 1 i m i points of C under (i, m −i). Note that the points (i, m −i) for 1 ≤ i ≤ m are not included. (m) e o The maps ψ arethendefinedinasimilarfashion to ψ , ψ . In the following, we use n n the notation r to stand for a wave of length r on C , a shift of the points of C enclosed (m) m m 48 Page 42 of 46 Balázs et al. Res Math Sci (2022) 9:48 (m) −m in the rectangle with opposite corners (1, −1), (r, −k). Given ω ∈ H , the map ψ stacks (ω − I{i ≡ m mod k}) copies of the wave i (whenever this is non-negative) −i (m) vertically in increasing order and then removes a point from the bottom of each of the columns 1, 2, ... ,i for each i ≡ m mod k such that ω = 0 (giving the generalised Young −i diagramofanelement of GFP (n)). See Fig. 14 for an example when k = 3(as D ,D ,−m k k earlier, points are labelled as black squares). It is possible to check that these maps are bijections, but as before this is tedious and is left to the reader. The cases for other offset can be proved in the usual fashion, by proving the equalities of generating functions: k(+1) −m f (q) = f (q)q if k = k − m with m ∈{0, 1, ... k − 1}. D ,D ,−m D ,D ,k k k k k These follow naturally from a generalisation of Wright’s bijection and the bijections e o φ , φ of Sect. 3.3, i.e. bijections ,n ,n k( + 1) (m) φ :GFP (n) → GFP n + − m D ,D ,−m D ,D ,k−m ,n k k k k for each  ∈ Z and m ∈{1, 2, ... ,k − 1} as follows. Given an element GFP (n), the D ,D ,−m k k (m) |k−m|(|k−m|+1) map φ adds the points inside the right angled triangle of size attached ,n 2 to either the left or top edge of its generalised Young diagram, depending on whether  ≥ 0 or < 0, respectively. It then uses the new leading diagonal implied by the triangle to k(+1) read off an element of GFP n + − m .See Fig. 15 for examples when D ,D ,k−m k k 2 k = 3. In general not much is known about the functions f (q) for arbitrary k and k . D ,D ,k k k As mentioned earlier, the functions  (q)(= f (q)) are studied in Andrews’ book D ,D ,0 k k [1], and we saw that  (q)and  (q) have product expansions (which gave us product 1 2 expansions for the specialised normalising factors). It turns out that  (q)alsohas a product expansion (proved in Corollary 5.1of[1], assuming Jacobi triple product) and so (3) S (q) can be written as: (3) S (q) =  (q) 2 3 4 5 6 7 8 = 1 + q + 3q + 6q + 11q + 18q + 31q + 49q + 78q + ··· 12i−6 (1 − q ) = . 6i−5 6i−4 2 6i−3 3 6i−2 2 6i−1 12i (1 − q )(1 − q ) (1 − q ) (1 − q ) (1 − q )(1 − q ) i≥1 As in the 2-exclusion case, it would be interesting to know if this could be proved using purely probabilistic methods. It would also be interesting to know whether the other functions f (q)and f (q) are products, as this would imply that the D ,D ,−1 D ,D ,−2 3 3 3 3 (3) (3) normalising factors S (q)and S (q) are products. −1 −2 For k ≥ 4, the functions  (q) are not expected to be products, but can be shown to be sums of products. However, the formulae are very tedious to write down and probably not too illuminating. We finish with another natural question. We have seen that the family of (non- degenerate) nearest neighbour interacting 0-1-2 systems on Z satisfying the blocking measure axioms can be stood up in a uniform way, leading to three variable Jacobi style identities. By specialising to the case of 2-exclusion, we then get two variable identities that are a special case of Theorem 1.3 (setting k = 2). Is there a family of 0-1-···-k processes Balázs et al. Res Math Sci (2022) 9:48 Page 43 of 46 48 ´ ´ MARTON BALAZS, DAN FRETWELL, AND JESSICA JAY (0) 2,23 −→ 42221 644431 0 0 33100 −−−−−− 11 (a) An element of GFP (23) and its image in GFP (32) D ,D ,0 D ,D ,6 3 3 3 3 (1) 1,23 −→ − 542000 76422 2 0 2 211000 −−−−− 00 (b) An element of GFP (23) and its image in GFP (30) D ,D ,−1 D ,D ,5 3 3 3 3 (2) 1,23 + −→ −− 53110 7533 2 1 0 2 2 21100 −−−− 000 (c) An element of GFP (23) and its image in GFP (28) D ,D ,−2 D ,D ,4 3 3 3 3 (0) (1) (2) Fig. 15 Bijections φ , φ and φ 2,23 1,23 1,23 for each k that is nicely behaved (i.e. has blocking measure and can be uniformly stood up) giving a multivariable identity that explains all cases of Theorem 1.3? In general we cannot expect the whole family of 0-1…-k systems on Z to be this well behaved (the stood up processes no longer have nearest neighbour interactions). However, there could be a low-dimensional subfamily of processes for each k that works. Acknowledgements The authors are grateful for discussions with János Engländer regarding the parity of the sum of Bernoulli random variables and for suggestions of an anonymous referee regarding the combinatorial background of our identities. M. Balázs was partially supported by the EPSRC EP/R021449/1 Standard Grant of the UK. On behalf of all authors, the corresponding author states that there is no conflict of interest and that there are no associated data for this manuscript. Appendix A The following are the non-empty sets GFP (n) for 0 ≤ n ≤ 8, whose sizes agree D ,D ,0,m 2 2 with the coefficients in the expansion of S (˜q, t) given in Sect. 3.3. even 48 Page 44 of 46 Balázs et al. Res Math Sci (2022) 9:48 GFP (0) = , D ,D ,0,0 2 2 GFP (1) = , D ,D ,0,2 2 2 GFP (2) = , D ,D ,0,0 2 2 1 0 GFP (2) = , , D ,D ,0,2 2 2 0 1 2 1 0 10 00 GFP (3) = , , , , , D ,D ,0,2 2 2 0 1 2 00 10 11 00 GFP (4) = , , D ,D ,0,0 2 2 00 11 3 2 1 0 20 00 GFP (4) = , , , , , , D2,D2,0,2 0 1 2 3 00 20 GFP (4) = , D ,D ,0,4 2 2 4 3 2 1 0 30 21 11 10 00 00 100 GFP (5) = , , , , , , , , , , , , D ,D ,0,2 2 2 0 1 2 3 4 00 00 10 11 21 30 100 20 10 GFP (5) = , , D ,D ,0,4 2 2 10 20 22 11 00 GFP (6) = , , , D ,D ,0,0 2 2 00 11 22 5 4 3 2 1 0 40 31 20 GFP (6) = , , , , , , , , , D ,D ,0,2 2 2 0 1 2 3 4 5 00 00 11 11 00 00 200 110 100 100 , , , , , , , 20 31 40 100 100 110 200 30 21 20 10 10 GFP (6) = , , , , , D ,D ,0,4 2 2 10 10 20 21 30 6 5 4 3 2 1 0 50 41 32 GFP (7) = , , , , , , , , , , D ,D ,0,2 2 2 0 1 2 3 4 5 6 00 00 00 22 30 21 11 11 10 00 00 , , , , , , , , 10 11 11 21 30 22 32 41 00 300 200 200 110 110 100 , , , , , , , 50 100 200 110 200 110 300 40 31 30 21 20 20 10 10 210 100 GFP (7) = , , , , , , , , , , D ,D ,0,4 2 2 10 10 20 20 21 30 31 40 100 210 33 22 11 00 1100 GFP (8) = , , , , , D ,D ,0,0 2 2 00 11 22 33 1100 7 6 5 4 3 2 1 0 60 51 42 40 GFP (8) = , , , , , , , , , , , , D ,D ,0,2 2 2 0 1 2 3 4 5 6 7 00 00 00 11 31 22 20 11 11 00 00 00 100 220 , , , , , , , , , , 11 20 22 31 40 42 51 60 400 100 211 100 300 200 300 110 100 400 , , , , , , , , 100 211 200 300 110 300 220 100 50 41 32 40 31 30 21 21 30 20 20 GFP (8) = , , , , , , , , , , D ,D ,0,4 2 2 10 10 10 20 20 30 30 21 21 31 40 10 10 10 310 210 210 110 200 100 , , , , , , , , . 32 41 50 100 200 110 210 210 310 The following are the non-empty sets GFP (n) for 0 ≤ n ≤ 8, whose sizes agree D ,D ,−1,m 2 2 with the coefficients in the expansion of S (˜q, t) given in Sect. 3.3. odd Balázs et al. Res Math Sci (2022) 9:48 Page 45 of 46 48 GFP (0) = , D ,D ,−1,1 2 2 − − 0 GFP (1) = , , D2,D2,−1,1 1 00 − − 1 GFP (2) = , , D ,D ,−1,1 2 2 2 00 − 0 GFP (2) = , D ,D ,−1,3 2 2 − − 2 − 0 −00 GFP (3) = , , , , D ,D ,−1,1 2 2 3 00 11 100 − 1 − 0 GFP (3) = , , D ,D ,−1,3 2 2 10 20 − − 3 − 1 −00 −00 GFP (4) = , , , , , D ,D ,−1,1 2 2 4 00 11 200 110 − 2 − 1 − 0 − 0 −10 GFP (4) = , , , , , D ,D ,−1,3 2 2 10 20 21 30 100 − − 4 − 2 − 0 −11 −00 GFP (5) = , , , , , , D ,D ,−1,1 2 2 5 00 11 22 100 300 − 3 − 2 − 1 − 1 − 0 − 0 GFP (5) = , , , , , , D ,D ,−1,3 2 2 10 20 21 30 31 40 −20 −10 −10 −00 , , , , 100 110 200 210 − − 5 − 3 − 1 −11 −11 GFP (6) = , , , , , , D ,D ,−1,1 2 2 6 00 11 22 110 200 −00 −00 −00 −100 , , , , 211 220 400 1100 − 4 − 3 − 2 − 2 − 1 − 1 − 0 − 0 GFP (6) = , , , , , , , , D2,D2,−1,3 10 20 21 30 31 40 32 41 − 0 −30 −21 −20 −20 −10 −00 , , , , , , , 50 100 100 110 200 300 310 −10 GFP (6) = , D ,D ,−1,5 2 2 − − 6 − 4 − 2 − 0 −22 −11 GFP (7) = , , , , , , , D ,D ,−1,1 2 2 7 00 11 22 33 100 300 −00 −00 −00 −200 −110 , , , , , 221 311 500 1100 1100 − 5 − 4 − 3 − 3 − 2 − 2 − 1 − 1 − 1 − 0 GFP (7) = , , , , , , , , , , D ,D ,−1,3 2 2 10 20 21 30 31 40 32 41 50 42 − 0 − 0 −40 −31 −30 −30 −21 −21 , , , , , , , , 51 60 100 100 110 200 110 200 −20 −11 −10 −10 −10 −00 −00 −100 , , , , , , , , 300 210 211 220 400 320 410 2100 −10 −20 GFP (7) = , , , D ,D ,−1,5 2 2 310 210 − − 7 − 5 − 3 − 1 −22 −22 −11 GFP (8) = , , , , , , , D ,D ,−1,1 2 2 8 00 11 22 33 110 200 211 −11 −11 −00 −00 −00 −300 −100 , , , , , , , 220 400 330 411 600 1100 2200 − 6 − 5 − 4 − 4 − 3 − 3 − 2 − 2 − 2 − 1 GFP (8) = , , , , , , , , , D2,D2,−1,3 10 20 21 30 31 40 32 41 50 42 48 Page 46 of 46 Balázs et al. Res Math Sci (2022) 9:48 − 1 − 1 − 0 − 0 − 0 − 0 −50 −41 −32 , , , , , , , , , 51 60 43 52 61 70 100 100 100 −40 −40 −31 −31 −30 −21 −20 −20 , , , , , , , , 110 200 110 200 300 300 211 220 −20 −11 −10 −10 −10 −00 −00 −00 , , , , , , , , 400 310 221 311 500 321 420 510 −210 −200 −110 −100 −100 , , , , . 1100 2100 2100 2 110 3100 −30 −21 −20 −10 −10 GFP (8) = , , , , . D ,D ,−1,5 2 2 210 210 310 320 410 Received: 12 June 2022 Accepted: 15 June 2022 Published online: 21 July 2022 References 1. Andrews, G.: Generalized Frobenius partitions. Am. Math. Soc. 49(301), 66 (1984) 2. Balázs, M.: Growth fluctuations in a class of deposition models. Ann. Inst. H. Poincaré Probab. Stat. 39(4), 639–685 (2003) 3. Balázs, M., Bowen, R.: Product blocking measures and a particle system proof of the Jacobi triple product. Ann. Inst. H. Poincaré Probab. Stat. 54(1), 514–528 (2018) 4. Carinci, G., Giardiná, C., Redig, F., Sasamoto, T.: A generalized asymmetric exclusion process with U (sl ) stochastic q 2 duality. Probab. Theory Relat. Fields 166, 887–933 (2016) 5. Engländer, J., Volkov, S.: Turning a coin over instead of tossing it. J. Theor. Probab. 31(2), 1097–1118 (2018) 6. Liggett, T.: Interacting Particle Systems. Classics in Mathematics, Sprigner, Berlin (2005) 7. Online Encyclopedia of Integer Sequences (A137829). https://oeis.org/A137829 8. Online Encyclopedia of Integer Sequences (A201077). https://oeis.org/A201077 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

Interacting particle systems and Jacobi style identities

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Abstract

jessica.jay@bristol.ac.uk School of Mathematics, Fry We consider the family of nearest neighbour interacting particle systems on Z allowing Building, University of Bristol, 0, 1 or 2 particles at a site. We parametrise a wide subfamily of processes exhibiting Woodland Road, Bristol BS8 1UG, UK product blocking measure and show how this family can be “stood up” in the sense of Balázs and Bowen (Ann Inst H Poincaré Probab Stat 54(1):514–528, 2018). By comparing measures, we prove new three variable Jacobi style identities, related to counting certain generalised Frobenius partitions with a 2-repetition condition. By specialising to specific processes, we produce two variable identities that are shown to relate to Jacobi triple product and various other identities of combinatorial significance. The family of k-exclusion processes for arbitrary k are also considered and are shown to give similar Jacobi style identities relating to counting generalised Frobenius partitions with a k-repetition condition. 1 Introduction The Jacobi triple product is a two variable identity relating an infinite sum with an infinite product: k k 2i 2i−1 2i−1 −1 q z = (1 − q )(1 + q z)(1 + q z ). i=1 k∈Z It is a foundational identity that has many applications in combinatorics and number theory, e.g. the theory of partitions and the theory of modular forms (Jacobi forms). While the Jacobi triple product is a classical identity and can be proved using elementary ad hoc methods, one finds that most modern proofs exhibit it as an equality of traces of operators acting on certain infinite-dimensional vector spaces. For example, algebraically it is well known that the identity manifests itself in the characters of certain infinite- dimensional representations of affine Lie algebras (e.g. it is the denominator identity for (1) sl ). In physics, it is well known as a consequence of the Boson–Fermion correspondence; both sides of the identity represent the same partition function, calculated in two ways by cataloguing fermionic and bosonic states by their energy and charge relative to a ground state (eigenvalues of operators acting on the two infinite-dimensional Fock spaces of states). © The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 0123456789().,–: volV 48 Page 2 of 46 Balázs et al. Res Math Sci (2022) 9:48 In [3], Balázs and Bowen found a probabilistic proof of the Jacobi Triple Product by interpreting it as an equality between the ASEP blocking measure (product Bernoulli) and the corresponding “stood up” AZRP measure (product geometric). The infinite sum arises via normalisation of the ASEP measure. The ASEP-AZRP correspondence can be seen as the probabilistic analogue of the Boson–Fermion correspondence (with the conserved quantity of a state relative to the zero site being the analogue of charge relative to a ground state). The advantage of the algebraic/physical interpretations of the identity is that they can be vastly generalised, e.g. by considering other representations, other Lie algebras, other Fock spaces. Each approach provides more Jacobi style identities, e.g. the Macdonald identities and the the Rogers–Ramanujan identities. These lead to other interesting connections with partitions and modular forms. Given this, it is natural to ask which other interacting particle systems provide Jacobi style identities. Of course not all processes will work, there are restrictions. For example, in order to get Jacobi style identities it seems necessary to find processes on Z that can be stood up and have interesting blocking measures (product or near product). The only such example for 0-1 processes is ASEP (as seen in [3]). In this paper, we parametrise a family of (non-degenerate) 0-1-2 systems on Z with product blocking measure, purely in terms of conditions on the rates. We then show that they are a valid family to consider in the above sense, giving precise details of stationary blocking measures and stood up processes. The corresponding measures then depend on three parameters q, ˜ t and c, and viewing these as variable over the family we prove the −c following three variable Jacobi style identities (letting z = q˜ ): Theorem 1.1 For 0 < q˜ < 1,t ≥ 1 and z > 0 (+1) 2 i 2 2i −1 i−1 −2 2(i−1) 2 S (˜q, t)˜q z = (1 + tzq˜ + z q˜ )(1 + tz q˜ + z q˜ ) even ∈Z i≥1 i 2 2i −1 i−1 −2 2(i−1) + (1 − tzq˜ + z q˜ )(1 − tz q˜ + z q˜ ) i≥1 (+1) 2+1 i 2 2i −1 i−1 −2 2(i−1) 2t S (˜q, t)˜q z = (1 + tzq˜ + z q˜ )(1 + tz q˜ + z q˜ ) odd ∈Z i≥1 i 2 2i −1 i−1 −2 2(i−1) − (1 − tzq˜ + z q˜ )(1 − tz q˜ + z q˜ ), i≥1 The functions S (˜q, t)and S (˜q, t) are normalising factors for measures on the two even odd irreducible components of the stood up process: 2 I{ω ≥1}− I{ω =0} iω + i(ω −1) −i −i −i −i i odd i even i odd i even S (˜q, t) = q˜ t , even ω∈H 2 I{ω ≥1}− I{ω =0} iω + i(ω −1) −i −i −i −i i even i odd i even i odd S (˜q, t) = q˜ t odd ω∈H Here the state space H is the set of sequences (ω ) of non-negative integers with no −i i≥1 two consecutive zeroes and agreeing with the sequence (0, 1, 0, 1, ... , ) far enough to the right. The state space H is similar but agrees with (1, 0, 1, 0, ...) for enough to the right. Balázs et al. Res Math Sci (2022) 9:48 Page 3 of 46 48 The normalising factors S (˜q, t)and S (˜q, t) as written above might look unappeal- even odd ing (even though they do count explicit combinatorial objects). The form presented here has direct motivation from the standing up procedure of the 0-1-2 state particle systems. In its proof, we demonstrate how this form came to life via such probabilistic ideas. We thank an anonymous referee for pointing out that a purely combinatorial proof is also possible, without the use of probability. We thought it is worth seeing the probability connection while proving Theorem 1.1, hence opted to keep our original argument. We also thank the anonymous referee for the challenge of finding natural alternative forms for S and S . We spent considerable effort finding a closed form of these even odd expressions. While we were not successful, an alternative probabilistic interpretation, giving rather different but still not closed formulas, is subject of a forthcoming paper. Nevertheless, numerical computation to high accuracy has led us to conjecture the fol- lowing equalities: Conjecture 1.2 n+i−1 n−i n n 2i (−1) q˜ t i≥1 n≥i 2i−1 i S (˜q, t) =  + even 2m m 2 (1 − q˜ ) (1 − q˜ ) m≥1 m≥1 n+i 2n+1 2m 3 n−i n(n+1) 2i (−1) q˜ t (1 − q˜ ) m≥1 i≥1 n≥i 2i 2i+1 S (˜q, t) =  +  . odd m 2 m 2 (1 − q˜ ) (1 − q˜ ) m≥1 m≥1 The identities in Theorem 1.1 are shown to have combinatorial significance. By adapt- ing the “General Principle” found in Andrews’ book ( [1]), we find that the RHS of these identities relate to generating functions for generalised Frobenius partitions (GFP’s) satis- fying a 2-repetition condition on the rows and a condition on the number of non-repeated entries. The content of these identities is then that the normalising factors S (˜q, t)and even S (˜q, t) are really formal generating functions for such GFP’s. This is not clear from their odd explicit definitions, and we explain this in detail in Sect. 3.3. In Sect. 4, we specialise to specific 0-1-2 systems and recover various two variable identities, also of combinatorial significance. The ASEP(q, 1) process of Redig et al. (found in [4]) is an extension of classical ASEP to allow two particles. It gives identities relating directly to Jacobi triple product. A (3-state) asymmetric particle-antiparticle exclusion process is considered and gives identities relating directly to the square of Jacobi triple product (which is also seen to have a combinatorial interpretation in terms of 2-coloured GFP’s). The 2-exclusion process gives identities relating to GFP’s with only a 2-repetition condition on the rows. Finally, in Sect. 5 we generalise the last example to the entire family of k-exclusion processes on Z. These are not 0-1-2 systems and so we have to adapt our techniques further. However, they are sufficiently nicely behaved to allow product blocking measure and have stood up processes, which we describe in detail. The identities we recover are as follows. Theorem 1.3 For 0 < q < 1,z = 0 and m ∈{0, 1, ... ,k − 1} k(+1) (k) −m k−m k S (q)q z −m ∈Z 48 Page 4 of 46 Balázs et al. Res Math Sci (2022) 9:48 ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ k−1 k k −rm −αr αi α αr α(i−1) −α ⎝ ⎝ ⎠ ⎝ ⎠ ⎠ = ζ ζ q z ζ q z . k k k r=0 i≥1 α=0 α=0 (k) The functions S (q) for m ∈{0, 1, ... ,k − 1} are normalising factors for measures on −m the k irreducible components of the stood up process: iω + i(ω −1) −i −i (k) i∈ /kZ+m i∈kZ+m S (q) = q . −m −m ω∈H −m Here the state space H is the set of sequences (ω ) of non-negative integers that −i i≥1 have no k consecutive zeroes and satisfy ω = I{i ≡ m mod k} for large enough i (in −i e o analogy with H and H ). The “General Principle” once again explains the combinatorial (k) nature of these identities. The content is that the normalising factors S (q) are formal −m generating functions for GFP’s with k-repetition condition in the rows (functions studied in detail in Andrews’ book [1]). Throughout the paper, we discuss all of the above in both probabilistic and combinatorial detail, giving full justification where possible and otherwise outlining the mysteries implied by one approach to the other. 2 Interacting particle systems with product blocking measure We recall the family of particle systems with product blocking measure introduced by Balázs and Bowen in [3]. For possibly infinite integers, −∞ ≤  ≤ 0 ≤ r ≤∞,wedefine :={i :  − 1 < i < r + 1}⊆ Z, and for two other possibly infinite integers −∞ ≤ min max min max ω ≤ 0 <ω ≤∞,wedefine I :={z : ω − 1 < z <ω + 1}⊆ Z.Weconsider interacting particle systems on the state space  ={η ∈ I :(> −∞ or N (η) < ∞)and (r < ∞ or N (η) < ∞)}, where N ,N : I → Z ∪{∞} are defined by h p h ≥0 0 r min max N (η) = (η − ω )and N (η) = (ω − η ) p i h i i= i=1 min max (when ω = 0and ω < ∞ this is just the number of particles to the left and holes 1 (i,j) to right of ). Given a state η ∈ , the state η ∈  obtained by a particle jumping from site i to j is given by η if k = i, j, (i,j) η = η −1if k = i, k ⎪ η +1if k = j. We will only consider processes with nearest neighbour interactions, i.e. the only states (i,i+1) (i+1,i) we can reach from η are η or η . The system then evolves according to Markov generators; in the ‘bulk’, i.e.  ≤ i ≤ r − 1 the generator has the form r−1 bulk (i,i+1) (i+1,i) L ϕ (η) = p(η , η ) ϕ(η ) − ϕ(η) + q(η , η ) ϕ(η ) − ϕ(η) i i+1 i i+1 i= for some cylinder function ϕ :  → R and functions p, q : I → [0, ∞)(theright andleft jump rates, respectively). If > −∞, we consider an open left boundary with boundary (−1,) jump rates p ,q : I → [0, ∞). We introduce the notation η to denote the state reached from η by a particle entering the system through the left boundary and similarly Balázs et al. Res Math Sci (2022) 9:48 Page 5 of 46 48 (,−1) η to be the state where a particle has left through the boundary. So the left boundary generator is of the form (−1,) (,−1) L ϕ (η) = p (η ) ϕ(η ) − ϕ(η) + q (η ) ϕ(η ) − ϕ(η) . Similarly if r < ∞, we consider an open right boundary with boundary jump rates p ,q : r r (r+1,r) I → [0, ∞). We let η denote the state reached from η when a particle enters the (r,r+1) system through the right boundary and similarly η the state where a particle has left through the boundary. So the right boundary generator is of the form r (r,r+1) (r+1,r) L ϕ (η) = p (η ) ϕ(η ) − ϕ(η) + q (η ) ϕ(η ) − ϕ(η) . r r r r In order to be a member of the blocking family, the jump rates of the system must obey the following conditions: min max (B1) For −∞ <ω <ω < ∞ we have that min max max min p(ω , ·) = p(·, ω ) = q(ω , ·) = q(·, ω ) = 0. If > −∞, min max q (ω ) = p (ω ) = 0 and if r < ∞, max min q (ω ) = p (ω ) = 0. r r (B2) The system is attractive, that is, p(·, ·) is non-decreasing in the first variable and non- increasing in the second whilst q(·, ·) is non-increasing in the first variable and non- decreasing in the second. If > −∞, p is non-increasing and q non-decreasing. If r < ∞, p is non-decreasing and q non-increasing. r r (B3) There exist p ,q ∈ R satisfying < p = 1 − q ≤ 1, and functions asym asym asym asym f : I → [0, ∞)and s : I × I → [0, ∞) such that p(y, z) = p · s(y, z + 1) · f (y)and q(y, z) = q · s(y + 1,z) · f (z). asym asym min min max If ω is finite, then f (ω ) = 0, and if ω is finite, we extend the domain of s max max and require that s(ω + 1, ·) = s(·, ω + 1) = 0. (Note that attractivity implies that s is non-increasing in both of its variables and f is non-decreasing). A priori the above definition allows many of the rates to be zero. However, in this paper we will assume that all rates, except those in (B1), are non-zero. Roughly speaking this min allows us to ignore degenerate processes and assume that f (z) > 0if z >ω (which we will do from now on). In general, there are many stationary distributions for systems satisfying the above. In this paper, we will be interested in the following one-parameter family of product stationary blocking measures, written explicitly in terms of p , q and f . asym asym Theorem 2.1 ([3], Theorem 3.1) For each c ∈ R, there is a product stationary blocking measure μ on , given by the marginals (i−c)z asym 1 q asym μ (z) = for i ∈  and z ∈ I, Z f (z)! i 48 Page 6 of 46 Balázs et al. Res Math Sci (2022) 9:48 ⎪ f (y) for z > 0 y=1 1 for z = 0 where Z is the normalising factor and f (z)! := ⎪ for z < 0. ⎪ 0 f (y) y=z+1 −1 Remark It is clear that if f and s satisfy (B3), then so do αf and α s for any α> 0. This scaling simply shifts the value of c in the blocking measure (i−c)z (i−c˜)z p p asym asym 1 q 1 q asym asym c c˜ μ (z) = = = μ (z). i i Z α f (z)! Z f (z)! i i max min max So without loss of generality we can assume that f (y) = 1 when ω and ω min y=ω +1 are finite. As expected, being a member of the blocking family imposes strict constraints on the min jump rates. In particular, for each y, z ∈ I \{ω } we can use (B3) to write f (z) p q(y − 1,z) asym = . f (y) q p(y, z − 1) asym Setting y = z and recalling that p = 1 − q gives asym asym p(y, y − 1) q(y − 1,y) p = and q = . asym asym q(y − 1,y) + p(y, y − 1) q(y − 1,y) + p(y, y − 1) min Note that p > q implies that p(y, y − 1) > q(y − 1,y) for all y ∈ I \{ω }.This, asym asym along with (B2), shows the condition min max (a) p(y, z) > q(z, y) for all y ∈ I \{ω } and z ∈ I \{ω } (i.e. the process is asymmetric with right drift). 1 1 Also by assumption < p is a constant (as is q < ) and so we must have the asym asym 2 2 condition that p(y, y − 1) min (b) = constant > 1 for all y ∈ I \{ω }. q(y − 1,y) min We then see that for y, z ∈ I \{ω } f (z) p(z, z − 1) q(y − 1,z) = . f (y) q(z − 1,z) p(y, z − 1) f (y) f (z) Since · = 1, this gives the condition f (z) f (y) p(z, z − 1)p(y, y − 1)q(y − 1,z)q(z − 1,y) min (c) = 1 for all y, z ∈ I \{ω }. q(z − 1,z)q(y − 1,y)p(y, z − 1)p(z, y − 1) min It is then clear that the function s is uniquely determined as follows, for y, z ∈ I \{ω } p(y, z − 1) q(y − 1,y) + p(y, y − 1) q(y − 1,z) q(y − 1,y) + p(y, y − 1) ( ) ( ) s(y, z) = = . f (y)p(y, y − 1) f (z)q(y − 1,y) To summarise, conditions (a), (b) and (c) are necessary conditions on the rates that are implied by being a member of the blocking family. However, in general a process with rates only satisfying (B1), (B2) and these three conditions is not expected to be a member of the blocking family (the quantities p and q as written above are well defined asym asym and satisfy < p = 1 − q ≤ 1, but it is not clear that the functions f and s should asym asym exist purely from these conditions). Balázs et al. Res Math Sci (2022) 9:48 Page 7 of 46 48 3 General 0-1-2 systems on Z with blocking measure For the choices I ={0, 1} and  = Z, the only member of the blocking family is ASEP, handled in [3]. In this section, we will consider the case of I ={0, 1, 2} and  = Z. In this case, the conditions (B1), (B2), (B3) translate into the following conditions on the rates (using the assumptions and discussions in Sect. 2): (B1) p(0, ·) = p(·, 2) = q(2, ·) = q(·, 0) = 0, (B2) p(2, ·) ≥ p(1, ·) > 0, p(·, 0) ≥ p(·, 1) > 0, q(0, ·) ≥ q(1, ·) > 0, q(·, 2) ≥ q(·, 1) > 0, (B3) For all y, z ∈{0, 1, 2},wehave p(y, z) = p · s(y, z + 1) · f (y)and q(y, z) = q · s(y + 1,z) · f (z) asym asym with p ,q given by asym asym p(1, 0) 1 q(0, 1) 1 p = > q = < asym asym q(0, 1) + p(1, 0) 2 q(0, 1) + p(1, 0) 2 and f, s given by ⎪ t (q(0, 1) + p(1, 0)) if y = z = 1, ⎧ ⎪ ⎪ tp(1,1)(q(0,1)+p(1,0)) ⎪ ⎪ 0if z = 0, if y = 1and z = 2, ⎪ ⎪ p(1,0) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ p(2,0)(q(0,1)+p(1,0)) −1 f (z):= s(y, z):= t if z = 1, if y = 2and z = 1, p(1,0)t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ p(2,1)(q(0,1)+p(1,0)) t if z = 2, if y = z = 2, p(1,0)t 0if y = 3or z = 3, for some t ≥ 1 (recall that we can assume f (1)f (2) = 1 without loss of generality). f (2) p(1,0)q(0,2) p(1,0)q(0,2) Remark Note that since = t = ,wesee that t = ≥ 1is f (1) q(0,1)p(1,1) q(0,1)p(1,1) uniquely determined. As previously mentioned, (B1), (B2) and (B3) imply the following necessary conditions for the rates: (a) p(y, z) > q(z, y) for all y ∈{1, 2} and z ∈{0, 1}, p(1,0) p(2,1) (b) = , q(0,1) q(1,2) p(1,0)p(2,1)q(1,1)q(0,2) (c) = 1. q(0,1)q(1,2)p(2,0)p(1,1) 48 Page 8 of 46 Balázs et al. Res Math Sci (2022) 9:48 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 e o (a) The “even” ground state, η (b) The “odd” ground state, η e o Fig. 1 Two ground states, η and η ,of Unlike the general case, these conditions are now also sufficient, in the sense that any process with I ={0, 1, 2} and  = Z satisfying (B1), (B2) and conditions (a), (b)and(c)will automatically satisfy (B3) with p ,q ,f and s as given above (all well defined). Thus, asym asym we have fully parametrised the family of such blocking processes using only conditions on the rates. By Theorem 2.1, for each such process there is a one parameter family of product stationary blocking measures μ given by the marginals I{z=1} −(i−c)z t q˜ μ (z) = for z ∈{0, 1, 2}. i c Z (˜q, t) asym q(0,1) c −(i−c) −2(i−c) Here 0 < q˜ = = < 1and Z (˜q, t):= 1 + tq˜ + q˜ (the normalising p i p(1,0) asym factor). Explicitly: ∞ 0 ∞ I{η =1} −(i−c)η I{η =1} (2−η )(i−c) i i i i t q˜ t q˜ c c μ (η) = μ (η ) = . i c c 2(i−c) Z (˜q, t) q˜ Z (˜q, t) i i i=−∞ i=−∞ i=1 Remark Note that these measures only depend on the parameter c and the quantities q, ˜ t attached to the process. Roughly speaking this will be the reason for getting a three variable identity later; as we run through the whole family of such processes, these parameters will be variables. Specialising to particular subfamilies of processes, for example, fixing t or letting t and q˜ be related will give two variable identities (c will still be a variable as will q˜, since the rates will typically depend on an asymmetry parameter 0 < q < 1). By definition of the state space, each η ∈  has η = 0 for all i small enough and η = 2 i i for all i big enough. We refer to these events as having a left most particle (LMP) and right most hole (RMH). By asymmetry, it then follows that the ground states of  (i.e. the most probable states) are all shifts of the following “even” and “odd” ground states (as in Fig. 1): 2if i ≥ 1, ⎨ ⎨ 2if i ≥ 1, e o η = η = 1if i = 0, i i ⎩ ⎪ 0 otherwise, ⎪ 0 otherwise. The reason for the terms “odd” and “even” will become clear in the next section. 3.1 Ergodic decomposition of ∞ 0 For any η ∈ , the quantity N(η):= (2 − η ) − η (as defined in [3]) is finite i i i=1 i=−∞ and is conserved by the dynamics of the process. So we can decompose  =  , n∈Z into irreducible components  :={η ∈  : N(η) = n}. Note that the left shift operator τ, n n−2 n defined by (τη) = η , gives a bijection  − →  (i.e. if η ∈  , then N(τη) = n − 2). i i+1 Balázs et al. Res Math Sci (2022) 9:48 Page 9 of 46 48 3 5 9 11 1 2 4 6 7 8 10 LMP 0 RMH -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 −1 (A) A state η ∈ Ω . (B) The stood up state ω. Fig. 2 An example of standing up −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 e o Fig. 3 “Even” and “odd” states ω and ω e e Remark Since N(η ) = 0, the shifts of η have even conserved quantity and give the n o ground states for the “even” part  of . Similarly, N(η ) =−1 and shifts provide n∈2Z n e o the ground states for the “odd” part  . This explains the labels η and η . n∈2Z+1 n,c c n We now calculate ν (·):= μ (·|N(·) = n), the unique stationary distribution on  . Lemma 3.1 The following relation holds c 2c−N(η) c μ (τη) = q˜ μ (η). This gives the recursion c n−2c c μ ({N = n}) = q˜ μ ({N = n − 2}). Proof (These are special cases of Lemma 6.1 and Corollary 6.2 in [3].) 0 ∞ I{η =1} −(i−c)η I{η =1} (2−η )(i−c) i+1 i+1  i+1 i+1 t q˜ t q˜ μ (τη) = c c 2(i−c) Z (˜q, t) q˜ Z (˜q, t) i=−∞ i=1 0 ∞ I{η =1} −(j−1−c)η I{η =1} (2−η )(j−1−c) j j j j t q˜ t q˜ j=−∞ j=1 2c = q˜ 2(i−c) q˜ Z (˜q, t) Z (˜q, t) i=1 i=−∞ 0 ∞ I{η =1} −(j−c)η I{η =1} (2−η )(j−c) j j j j 0 ∞ t q˜ t q˜ 2c+ η − (2−η ) j j j=−∞ j=1 j=−∞ j=1 = q˜ 2(i−c) q˜ Z (˜q, t) Z (˜q, t) i=1 i=−∞ 2c−N(η) c = q˜ μ (η). Since N(τη) = N(η) − 2, we have that c c μ ({N = n − 2}) = μ (η) η:N(η)=n−2 = μ (τ ηˆ) ηˆ:N(τ ηˆ)=n−2 2c−N(ˆ η) = q˜ μ (ˆ η) ηˆ:N(ˆ η)=n 48 Page 10 of 46 Balázs et al. Res Math Sci (2022) 9:48 2c−n c = q˜ μ ({N = n}). The general solution to the recursion above is n(n+2) ⎨ −nc c q˜ μ ({N = 0})if n ∈ 2Z, μ ({N = n}) = 2 (n+1) ⎩ −(n+1)c c q˜ μ ({N =−1})if n ∈ 2Z + 1. Since there is a dependence on parity, we will need to calculate the probability μ ({N(η) ∈ n,c 2Z + 1}) in order to finish our calculation of ν . Lemma 3.2 1 − 1 − 2μ (1) i=−∞ μ ({N(η) ∈ 2Z + 1}) = . Proof (We adapt the proof of [5], Proposition 1). Define the partial conserved quantity a 0 N (η) = (2 − η ) − η for a ≥ 1 a i i i=1 i=−a and note that N (η) → N(η)as a →∞. For each a ≥ 1, we define the random variables N (η) i=−a Y := (−1) = (−1) . Since η → 0as i →−∞ and η → 2as i →∞, we have that Y → Y as a →∞, where i i a 1if N(η) is even, N(η) Y := (−1) = −1if N(η) is odd. Since |Y |= 1 for all a ≥ 1, dominated convergence applies, and by the product structure of μ we have that ∞ ∞ c c c c c E [Y ] = lim E [Y ] = 1 · μ (0, 2) + (−1) · μ (1) = 1 − 2μ (1) , i i i a→∞ i=−∞ i=−∞ c c where E denotes the expectation taken w.r.t. μ . On the other hand, we have c c c E [Y ] = 1 · μ {N(η) ∈ 2Z} + (−1) · μ {N(η) ∈ 2Z + 1} = 1 − 2μ {N(η) ∈ 2Z + 1} . Thus, 1 − 1 − 2μ (1) i=−∞ μ ({N(η) ∈ 2Z + 1}) = and also 1 + 1 − 2μ (1) i=−∞ c c μ ({N(η) ∈ 2Z}) = 1 − μ ({N(η) ∈ 2Z + 1}) = . Balázs et al. Res Math Sci (2022) 9:48 Page 11 of 46 48 Now by combining Lemma 3.1 and Lemma 3.2, we get the unique stationary distribution n,c n ν on  . Proposition 3.3 For n ∈ 2Z, the unique stationary distribution on  is given by (+1)−2c c 2 q˜ μ (η)I{N(η) = n} =−∞ n,c ν (η) = . n(n+2) −nc q˜ 1 + 1 − 2μ (1) i=−∞ For n ∈ 2Z + 1,itisgiven by (+1) −2(+1)c c 2 q˜ μ (η)I{N(η) = n} =−∞ n,c ν (η) = . 2 ∞ (n+1) −(n+1)c q˜ 1 − 1 − 2μ (1) i=−∞ −(i−c) −2(i−c) 1−tq˜ +q˜ c c −(i−c) Notice that (1 − 2μ (1)) = , and so if we let W (˜q, t):= 1 − tq˜ + i i Z (˜q,t) −2(i−c) n,c q˜ ,wecan write ν as ∞ 0 ∞ (+1)−2c I{η =1} −η (i−c) I{η =1} (2−η )(i−c) i i i i 2 q˜ t q˜ t q˜ =−∞ i=−∞ i=1 n,c ν (η) = ∞ ∞ n(n+2) −nc c c c c 2(i−c) 2(i−c) q˜ q˜ Z (˜q, t)Z (˜q, t) + q˜ W (˜q, t)W (˜q, t) i −i+1 i −i+1 i=1 i=1 I{N(η) = n} when n is even and ∞ 0 ∞ (+1) −2(+1)c I{η =1} −η (i−c) I{η =1} (2−η )(i−c) i i i i 2 q˜ t q˜ t q˜ =−∞ i=−∞ i=1 n,c ν (η) = ∞ ∞ (n+1) −(n+1)c c c c c 2(i−c) 2(i−c) q˜ q˜ Z (˜q, t)Z (˜q, t) − q˜ W (˜q, t)W (˜q, t) i −i+1 i −i+1 i=1 i=1 I{N(η) = n} when n is odd. Remark These distributions are independent of c, as discussed in [3]. However, later we will need to stress the dependence of both the numerator and denominator on c and so we will keep c in our notation. 3.2 Standing up/laying down In this section, we transfer the dynamics on  to that of a restricted particle system on <0 Z , in direct analogy with the “standing up/laying down” method of [3]. By doing this, we ≥0 obtain an alternative characterisation of the stationary distributions given in Proposition 3.3. n th Definition 3.4 Given η ∈  ,let S be the site of the r particle when reading left to n <0 right, bottom to top. The corresponding stood up state is then T (η) = ω ∈ Z ,with ≥0 ω = S − S .See Fig. 2 for example. −r r+1 r Remark We can extend this method to stand up blocking processes with I = {0, 1, 2, ... ,k} and  = Z for any k ∈ N. However, the corresponding particle systems are 48 Page 12 of 46 Balázs et al. Res Math Sci (2022) 9:48 not always guaranteed to have product stationary blocking measures (this is dependent on the jump rates of the original system). n <0 A priori the “standing up” map T is an injection into Z . However, since η ≤ 2 for all ≥0 i, the image of T lies in the restricted state space <0 H :={ω ∈ Z : ω = 0 ⇒ ω = 0, ∀i > 0}. −i −i−1 ≥0 Since η = 2 for i large ω must coincide far to the left with one of the following “even” or e o e “odd” states (dependent on the parity of n); ω and ω such that ω = I{i ∈ 2Z } and ≥1 −i ω = I{i ∈ 2Z + 1} (see Fig. 3). ≥0 −i e 0 e Remark Note that all shifts of the “even” ground state η ∈  standuptogive ω . o −1 o Similarly all shifts of η ∈  give ω . This is further justification for the notation. n e o e o We now see that the image of T lies in H := H ∪ H , where the disjoint sets H and H are defined as e  e H ={ω ∈ H : ∃N >0s.t ω = ω ∀i ≥ N }, −i −i o  o H ={ω ∈ H : ∃N >0s.t ω = ω ∀i ≥ N }. −i −i For ω ∈ H, the minimum value of N satisfying the above will be denoted by E(ω)or O(ω), e o dependent on whether ω ∈ H or ω ∈ H . ⎨ e H if n ∈ 2Z, n n Lemma 3.5 T ( ) = ⎩ o H if n ∈ 2Z + 1. Proof It suffices to show surjectivity of T for each n. e n If n ∈ 2Z and ω ∈ H , then we construct the state η ∈  having leftmost particle at E(ω)−1 n+E(ω)+I{E(ω)∈2Z+1} th the site S = − ω and r particle at site S = S + ω for 1 −i r r−1 1−r i=1 each r ≥ 2. o n Similarly for n ∈ 2Z + 1and ω ∈ H construct the state η ∈  with left most particle O(ω)−1 n+O(ω)+I{O(ω)∈2Z} th at site S = − ω and r particle at site S = S + ω for 1 −i r r−1 1−r i=1 each r ≥ 2. n n It is clear that in either case T (η) = ω and hence T is surjective. The inverse maps described in the above proof are referred to as the “laying down” maps. Using the “standing up” maps we define a particle system on H whose dynamics are inherited from those on . In particular right jumps in η correspond to right jumps in ω and similarly for left jumps. The explicit right/left jump rates are given in Table 1 for r ≥ 2. Note that the jump rates p (1, ·)and q (·, 1) can be 0, due to the no consecutive zeroes ω ω condition, a jump that would cause ω = ω = 0 for some r ≥ 2 is blocked. −r −r+1 Since the “stood up ” process is only defined on the negative half integer line we must consider what happens at the boundary site. We consider an open infinite type boundary, that is a reservoir of particles at “site 0” at which particles can enter or leave the system with the rates given in Table 2. Balázs et al. Res Math Sci (2022) 9:48 Page 13 of 46 48 Table 1 Jump rates p (ω , ω ) and q (ω , ω ), ω −r −r+1 ω −r −r+1 respectively, of the stood up process ω = 0 ω ≥ 1 −r+1 −r+1 ω =00 0 −r ω = 1 p(2, 1)I{ω = 0} p(1, 1)I{ω = 0} −r −r−1 −r−1 ω ≥ 2 p(2, 0) p(1, 0) −r ω = 0 ω = 1 ω ≥ 2 −r+1 −r+1 −r+1 ω =00 q(1, 2)I{ω = 0} q(0, 2) −r −r+2 ω ≥10 q(1, 1)I{ω = 0} q(0, 1) −r −r+2 Table 2 Boundary jump rates for the stood up process Rate into the boundary Rate out of the boundary ω =00 q(0, 2) −1 ω = 1 p(1, 1)I{ω = 0} q(0, 1) −1 −2 ω ≥ 2 p(1, 0) −1 We note that the dynamics at the boundary in ω correspond exactly to that of the LMP −1 in T (ω). ( ) To find the stationary distribution for the “stood up” process we first consider the ∗ <0 unrestricted process ω ∈ Z , i.e. the process described by the same jump rates as ω ≥0 but where the number of consecutive zeros is not restricted. It is clear to see that the unrestricted process is a member of the blocking family; (B1) and (B2) follow directly from the fact that the rates of the original process satisfy these conditions and similarly condition (B3) is satisfied by the constants p(1, 0) 1 q(0, 1) 1 ∗ ∗ p = > and q = < asym asym q(0, 1) + p(1, 0) 2 q(0, 1) + p(1, 0) 2 and the functions tp(2,1)(q(0,1)+p(1,0)) ⎪ if y = z = 1, ⎪ p(1,0) ⎧ ⎪ ⎪ ⎪ 0if z = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ tp(1,1)(q(0,1)+p(1,0)) ⎪ ⎪ ⎪ ⎪ if y = 1and z ≥ 2, ⎪ ⎪ p(1,0) ⎨ ⎨ ∗ ∗ −1 f (z):= s (y, z):= t if z = 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p(2,0)(q(0,1)+p(1,0)) ⎪ ⎪ ⎪ ⎪ if y ≥ 2and z = 1, ⎪ ⎪ p(1,0)t ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ t if z ≥ 2, ⎩ q(0,1)+p(1,0) if y, z ≥ 2, q(0,1)p(2,0) with t = > 1. p(1,0)q(1,1) By Theorem 2.1 we can find a one parameter family of product stationary blocking ∗,cˆ <0 measures π on Z with marginals given by ≥0 (2−z)I{z≥1} (i+cˆ)z t q˜ ∗,cˆ π (z) = for z ≥ 0and i ≥ 1 −i ∗,cˆ Z (˜q, t) −i ∗,cˆ (here Z (˜q, t) is the normalising factor). −i We fix the value of cˆ by considering reversibility over the boundary edge (−1, 0). Suppose ∗,cˆ that π satisfies detailed balance over this boundary edge: ∗,cˆ ∗,cˆ π (y) · “rate into the boundary” = π (y − 1) · “rate out of the boundary” for all y ≥ 1. −1 −1 We consider the following two cases 48 Page 14 of 46 Balázs et al. Res Math Sci (2022) 9:48 (1) If y = 1, then this gives ∗,cˆ ∗,cˆ π (1)p(1, 1) = π (0)q(0, 2) −1 ∗,cˆ π (1) q(0, 2) p(2, 0)q(0, 1) −1 ⇔ = = by condition (B3)(c) ∗,cˆ 2 p(1, 1) p(1, 0) q(1, 1) π (0) −1 1+cˆ 2 ⇔ tq˜ = t q˜ cˆ ⇔ q˜ = t. (2) If y ≥ 2, then this gives ∗,cˆ ∗,cˆ π (y)p(1, 0) = π (y − 1)q(0, 1) −1 ∗,cˆ π (y) q(0, 1) −1 ⇔ = = q˜ ∗,cˆ p(1, 0) π (y − 1) −1 −1 1+cˆ ⇔ t q˜ = q˜ cˆ ⇔ q˜ = t. cˆ Thus, we should choose cˆ so that q˜ = t. The marginals now become (dropping cˆ from the notation) iz 2I{z≥1} q˜ t π (z) = for z ≥ 0and i ≥ 1. −i ∗ Z (˜q, t) −i Now that we have the stationary distribution for the unrestricted process we consider e o the restriction to H and find the stationary measure. Recall that H = H ∪ H , and note e e o that H is the irreducible component of the “even” ground state ω and similarly H for the “odd” ground state ω . We define stationary measures on these irreducible components ∗ e e o o in terms of π , getting π on H and π on H . It seems natural to define these measures e ∗ e o ∗ o ∗ as π (·) = π (·|· ∈ H )and π (·) = π (·|· ∈ H ). However, w.r.t π the probability of being in either irreducible component is zero and so these quantities are undefined. To rectify this, we use the following formal reasoning, e ∗ e π (ω) = π (ω|ω ∈ H ) ∗ e π (ω )I{ω ∈ H } −i −i i≥1 π (ω ) −i −i i≥1 ω ∈H π (ω ) −i −i I{ω ∈ H } ∗ e π (ω ) −i −i i≥1 = . π (ω ) −i −i ∗ e π (ω ) −i −i ω ∈H i≥1 This is now a well defined distribution since far to the left any configuration ω ∈ H agrees with ω , forcing the products to be finite and the denominator to no longer be 0. We apply a similar reasoning for π . We then have e e o o φ (ω )I{ω ∈ H } φ (ω )I{ω ∈ H } −i −i −i −i i≥1 i≥1 e o π (ω) =   and π (ω) =   , e  o φ (ω ) φ (ω ) −i −i −i −i e  o ω ∈H i≥1 ω ∈H i≥1 Balázs et al. Res Math Sci (2022) 9:48 Page 15 of 46 48 ∗ ∗ π (ω ) π (ω ) −i −i e −i o −i where φ (ω ) = and φ (ω ) = , given explicitly as follows: ∗ e ∗ o −i −i −i −i π (ω ) π (ω ) −i −i −i −i iω 2I{ω ≥1} −i −i q˜ t i ∈ 2Z + 1, φ (ω ):= −i −i i(ω −1) −2I{ω =0} −i −i q˜ t i ∈ 2Z, i(ω −1) −2I{ω =0} −i −i q˜ t i ∈ 2Z + 1, φ (ω ):= −i −i iω 2I{ω ≥1} −i −i q˜ t i ∈ 2Z e o and so we get the following explicit formulae for π (ω)and π (ω). e o Proposition 3.6 The unique stationary measures on H and H are given by 2 I{ω ≥1}− I{ω =0} iω + i(ω −1) −i −i −i −i iodd ieven iodd ieven q˜ t π (ω) = , S (˜q, t) even 2 I{ω ≥1}− I{ω =0} iω + i(ω −1) −i −i −i −i ieven iodd ieven iodd q˜ t π (ω) = . S (˜q, t) odd e o Here S (˜q, t)and S (˜q, t) are normalising factors with respect to H and H : even odd 2 I{ω ≥1}− I{ω =0} iω + i(ω −1) −i −i −i −i i odd i even i odd i even S (˜q, t) = q˜ t , even ω ∈H 2 I{ω ≥1}− I{ω =0} iω + i(ω −1) −i −i −i −i i even i even i odd i odd S (˜q, t) = q˜ t . odd ω ∈H Proof It is well known that the restriction of a reversible stationary measure on a contin- uous time Markov process is also reversible stationary (see Proposition 5.10 of [6]). The o e ∗ result follows since π and π are restrictions of π . It would be interesting to know whether the normalising factors S (˜q, t)and S (˜q, t) even odd can be written as infinite products. We will see later that when specialising to certain processes (i.e. choosing certain values for q˜ and t) the specialised normalising factors appear to be products with combinatorial significance. We should compare the situation with that of [3], in which the authors stand up ASEP to get AZRP and get the partition 2i −1 function (1 − q ) as normalising factor (one of the components of the product i≥1 side of the Jacobi triple product). However, the fact that they naturally get a product seems to be a direct consequence of the choice of “standing up” map. Indeed it is possible to stand up ASEP in a different way, giving a normalising factor that has a similar form to the two given above (i.e. not naturally given as a product) but is then recognised as the above partition function. Unfortunately we do not see an obvious way to recognise the normalising factors S (˜q, t)and S (˜q, t) as products. even odd 48 Page 16 of 46 Balázs et al. Res Math Sci (2022) 9:48 3.3 Identities n n By Lemma 3.5, the standing up transformation T describes a bijection between e o n and one of the state spaces H or H , depending on the parity of n.Since T preserves n,c the dynamics of the corresponding processes, we get an equality of measures, ν (η) = e n n,c o n π (T (η)), when n ∈ 2Z and ν (η) = π (T (η)), when n ∈ 2Z + 1 (for all values of c). We will now see that evaluating these equalities at ground states leads to interesting combinatorial identities. Recall that a process on  with blocking measure has two ground states up to shift, e 0 o −1 0 e e −1 o o 0,c e e e η ∈  and η ∈  , satisfying T (η ) = ω and T (η ) = ω .Thus, ν (η ) = π (ω ) −1,c o o o and ν (η ) = π (ω ), and by Proposition 3.3 and Proposition 3.6 we get the following two identities (after rearrangement): (+1)−2c 2(i−c) c c 2 S (˜q, t)˜q = q˜ Z (˜q, t)Z (˜q, t) even i −i+1 =−∞ i≥1 2(i−c) c c + q˜ W (˜q, t)W (˜q, t), i −i+1 i≥1 (+1) −(2+1)c 2(i−c) c c 2t S (˜q, t)˜q = q˜ Z (˜q, t)Z (˜q, t) odd i −i+1 =−∞ i≥1 2(i−c) c c − q˜ W (˜q, t)W (˜q, t). i −i+1 i≥1 c c −c Writing Z (˜q, t)and W (˜q, t) explicitly and letting z = q˜ prove the following identities. i i Theorem 1.1 For 0 < q˜ < 1,t ≥ 1 and z > 0 (+1) 2 i 2 2i −1 i−1 −2 2(i−1) 2 S (˜q, t)˜q z = (1 + tzq˜ + z q˜ )(1 + tz q˜ + z q˜ ) even i≥1 ∈Z i 2 2i −1 i−1 −2 2(i−1) + (1 − tzq˜ + z q˜ )(1 − tz q˜ + z q˜ ) i≥1 (+1) 2+1 i 2 2i −1 i−1 −2 2(i−1) 2t S (˜q, t)˜q z = (1 + tzq˜ + z q˜ )(1 + tz q˜ + z q˜ ) odd ∈Z i≥1 i 2 2i −1 i−1 −2 2(i−1) − (1 − tzq˜ + z q˜ )(1 − tz q˜ + z q˜ ), i≥1 We were unable to find these three variable Jacobi style identities explicitly written down in the literature, and we believe that they are new. It is interesting to note that the above proof of these identities is purely probabilistic and makes no assumption of other classical identities; for example, Jacobi triple product (although we will see later that specialising to the ASEP(q, 1) process gives the odd/even parts of Jacobi triple product). We will now discuss the combinatorial nature of these identities. As is well known, the product (1 + q˜ ) is the generating function for partitions of n with distinct parts. In i≥1 i 2i a similar vein the product (1 + q˜ + q˜ ) is the generating function for partitions of i≥1 n with each part appearing at most twice. It is then clear that the two variable product i 2i n m (1 + tq˜ + q˜ ) = a q˜ t is the generating function for such partitions of n,m i≥1 n,m n with exactly m distinct parts. For example a = 3and a = 2 count the partitions 5,1 5,2 Balázs et al. Res Math Sci (2022) 9:48 Page 17 of 46 48 764 631 631 764 Fig. 4 Conjugate partitions 30 = 8 + 8 + 7 + 3 + 2 + 1 + 1and 30 = 7 + 5 + 4 + 3 + 3 + 3 + 3 + 2 with their corresponding GFP’s [5, 3 + 1 + 1, 2 + 2 + 1] and [4 + 1, 3 + 2], respectively (the partitions 2 + 1 + 1 + 1and 1 + 1 + 1 + 1 + 1 are not counted since 1 appears more than twice). Going one step further i 2 2i n m k the three variable product (1 + tzq˜ + z q˜ ) = a q˜ t z is the generating n,m,k i≥1 n,m,k function for such partitions of n which have exactly k parts in total. For example a = 5,1,1 1,a = 2and a = 2 count the partitions [5], [3 + 1 + 1, 2 + 2 + 1] and [4 + 1, 3 + 2], 5,1,3 5,2,2 i 2 2i −1 i−1 −2 2(i−1) respectively. The meaning of the product (1 +tzq˜ +z q˜ )(1 +tz q˜ +z q˜ ) i≥1 is more subtle and relates to generalised Frobenius partitions. A generalised Frobenius partition (GFP) of n is a two row array of integers a a ... a 1 2 s b b ... b 1 2 s such that a ≥ a ≥ ··· ≥ a ≥ 0, b ≥ b ≥ ··· ≥ b ≥ 0and s + (a + b ) = n. 1 2 s 1 2 s i i 1≤i≤s Given an ordinary partition of n we can produce a GFP of n by letting s be the length of the leading diagonal in the Young diagram, the a be the lengths of rows to the right of the diagonal and the b be the lengths of the columns under the diagonal (see Fig. 4). This map gives a bijection between ordinary partitions of n and GFP’s of n with each row having distinct entries (note that GFP’s in general allow repeats in the rows). Taking the conjugate of a partition becomes the natural operation of swapping rows in the corresponding GFP. This convenience was the classical motivation, but GFP’s are now studied as combinatorial objects in their own right. Naturally, we wish to count certain families of GFP’s. A “General Principle” due to n k Andrews, given in [1], provides a way to do this. Suppose f (˜q, z) = a q˜ z and n,k n,k n k f (˜q, z) = b q˜ z are generating functions for ordinary partitions of n with k parts n,k n,k and satisfying some conditions A and B, respectively. Then, the “General Principle” states −1 k that the formal series f (˜q, qz ˜ )f (˜q, z ) = f (˜q)z has constant term f (˜q) equal A B A,B,k A,B,0 to the generating function for GFP’s of n with first row satisfying condition A and second row satisfying condition B. The set of such GFP’s will be denoted GFP (n). We can A,B actually give a uniform interpretation of all of the formal series f (˜q) if we generalise A,B,k further to allow GFP’s having rows of unequal length, i.e. two row arrays of integers a a ... a 1 2 s b b ... b 1 2 s such that a ≥ a ≥ ··· ≥ a ≥ 0,b ≥ b ≥ ··· ≥ b ≥ 0and s + a + 1 2 s 1 2 s 1 i 1 2 1≤i≤s b = n. We will refer to these as GFP’s with offset s − s (so that GFP’s with i 1 2 1≤i≤s 2 48 Page 18 of 46 Balázs et al. Res Math Sci (2022) 9:48 offset 0 are classical GFP’s). The |s − s | “empty” entries in the shorter row will be 1 2 labelled to the left with a dash (this will be important later). The full power of the “General Principle” is then that the formal series f (˜q) is the generating function for the sets A,B,k GFP (n), defined as above but for (possibly non-zero) offset k. A popular choice of A,B,k condition on the rows is A = B = D , the condition that each part appears at most r times. One can use the “General Principle” to give a combinatorial interpretation of the famous Jacobi Triple Product identity (written here in an equivalent form to the one given in the introduction): k(k+1) i i i−1 −1 k (1 − q˜ )(1 + q˜ z)(1 + q˜ z ) = q˜ z . i≥1 k∈Z Indeed, rearranging gives: k(k+1) i i−1 −1 k (1 + q˜ z)(1 + q˜ z ) =  q˜ z (1 − q˜ ) i≥1 i≥1 k∈Z and so by applying the “General Principle” to the LHS we see that this identity is equivalent k(k+1) to the equalities f (˜q) = q˜ for all k ∈ Z. This is true for the base case D ,D ,k i 1 1 (1−q˜ ) i≥1 k = 0 since we have already seen that GFP (n) is in bijection with ordinary partitions D ,D ,0 1 1 of n. For other values of k the corresponding equality follows from the equivalence of k(k+1) generating functions f (˜q) = f (˜q)˜q , proved by the following bijection φ : D ,D ,k D ,D ,0 k 1 1 1 1 k(k+1) GFP (n) → GFP n + often attributed to Wright. Given an element D ,D ,0 D ,D ,k 1 1 1 1 2 of GFP (n) we have a corresponding ordinary partition of n. Adjoin a right angled D ,D ,0 1 1 |k|(|k|+1) triangle of size to either the left or top edge of its Young diagram, depending on whether k ≥ 0or k < 0, respectively (see Fig. 5). Then, use the new leading diagonal k(k+1) implied by the triangle to read off an element of GFP n + by letting s D ,D ,k 1 1 2 be the length of the diagonal if k ≥ 0(s if k < 0), the a be the sizes of rows to the 2 i right of the diagonal and the b be the sizes of columns under the diagonal (the |k| empty rows/columns coming from the triangle supply the required |k| “empty” entries in the corresponding GFP of offset k). In comparison, we can now give a combinatorial interpretation of the identities in Theorem 1.1. By a three variable adaptation of the “General Principle” the formal series: i 2 2i −1 i−1 −2 2(i−1) k (1 + tzq˜ + z q˜ )(1 + tz q˜ + z q˜ ) = f (˜q, t)z D ,D ,k 2 2 i≥1 k∈Z n m has coefficients f (˜q, t) = c q˜ t that are two variable generating functions D ,D ,k n,m,k 2 2 n,m for the sets GFP (n) ⊆ GFP (n), consisting of such GFP’s having a total of D ,D ,k,m D ,D ,k 2 2 2 2 m distinct parts (each row treated separately). The identities of Theorem 1.1 are then equivalent to the equalities: ⎨ (+1) S (˜q, t)˜q if k = 2, even f (˜q, t) = D ,D ,k 2 2 2 (+1) tS (˜q, t)˜q if k = 2 + 1. odd (The reason for various sign changes in the two identities is merely to separate the cases of even and odd offset, since these behave differently.) We have of course proved these equalities probabilistically, but it is not explicitly clear that the normalising factors are Balázs et al. Res Math Sci (2022) 9:48 Page 19 of 46 48 −→ 764 11 10 8 3 1 631 −− − − 2 (a) An element of GFP (30) and its image in GFP (40) D ,D ,0 D ,D ,4 1 1 1 1 −4 −→ 764 −− − − 320 631 10 7 5 3 210 (b) An element of GFP (30) and its image in GFP (36) D ,D ,0 D ,D ,−4 1 1 1 1 Fig. 5 Wright bijections φ and φ (cont) 4 −4 related to counting GFP’s with the 2-repetition condition. However, we are able to give combinatorial proofs of these equalities, similar to the case of Jacobi triple product. We start with the base cases k = 0and −1, i.e. that S (˜q, t) = f (˜q, t) even D ,D ,0 2 2 tS (˜q, t) = f (˜q, t). D ,D ,−1 odd 2 2 Using MAGMA, we were able to compute that the first few coefficients of the normalising factors are given by: 2 2 2 3 2 4 2 4 5 2 4 S (˜q, t) = 1 + qt ˜ + q˜ (1 + 2t ) + 5˜q t + q˜ (2 + 6t + t ) + q˜ (12t + 2t ) even 6 2 4 7 2 4 8 2 4 + q˜ (3 + 16t + 5t ) + q˜ (25t + 10t ) + q˜ (5 + 30t + 20t ) + ··· 2 2 3 2 4 2 5 2 S (˜q, t) = 1 + 2˜q + q˜ (2 + t ) + q˜ (4 + 2t ) + q˜ (5 + 5t ) + q˜ (6 + 10t ) odd 6 2 4 7 2 4 8 2 4 + q˜ (10+15t +t )+q˜ (12 + 26t + 2t ) + q˜ (15 + 40t + 5t ) + ··· (see Appendix for the lists of GFP’s of offset 0 and −1 that are counted by these coef- ficients.) Note that in both cases the exponents of t are even. This is expected since − 422 2 200 Fig. 6 Generalised Young diagrams corresponding to two GFP’s of 15 with offset 0 and −1 48 Page 20 of 46 Balázs et al. Res Math Sci (2022) 9:48 GFP (n) is empty when m ≡ k mod 2 (this justifies the extra t in the odd case). D ,D ,k,m 2 2 These expansions will be useful later when specialising to particular processes. In order to prove these base cases we seek analogues of the Frobenius bijection between ordinary partitions of n and elements of GFP (n). However, to describe these we must first D ,D ,0 1 1 generalise the notion of Young diagram to allow GFP’s with the 2-repetition condition. Elements of GFP (n) do not correspond to ordinary partitions and so do not D ,D ,0 2 2 naturally give rise to Young diagrams. However, they do naturally give rise to cer- tain finite subsets of C ={(n ,n ) ∈ Z : n + n ≡ 0 mod 2}. The subset cor- e 1 2 1 2 responding to such a GFP with s columns consists of the s leading diagonal points (1, −1), ... , (s, −s), the first a points of C to the right of (i, −i) and the first b points of C i e i e under (i, −i). Similarly, elements of GFP (n) will give well defined finite subsets of D ,D ,−1 2 2 C ={(n ,n ) ∈ Z : n + n ≡ 1 mod 2}. The subset corresponding to such a GFP with o 1 2 1 2 s entries on the top row contains the s leading diagonal points (2, −1), ... , (s + 1, −s ), 1 1 1 1 the first a points of C to the right of (i + 1, −i) and the first b points of C under i o i o (i, −i + 1) (the point (1, 0) is not included). See Fig. 6 for an example of each kind of generalised Young diagram (points are labelled by black squares; the white squares are included only for aesthetics). n m We can now return to proving the base cases. By comparing coefficients of q˜ t on both sides it suffices to find bijections: iω + i(ω − 1) = n e e −i −i i odd i even ψ : ω ∈ H : → GFP (n), D ,D ,0,m n,m 2 2 2 I{ω ≥ 1}− I{ω = 0} = m −i −i i odd i even iω + i(ω − 1) = n o o −i −i i even i odd ψ : ω ∈ H : → GFP (n). D ,D ,−1,m+1 n,m 2 2 2 I{ω ≥ 1}− I{ω = 0} = m −i −i i even i odd These maps can be constructed as restrictions of bijections: e e ψ : ω ∈ H : iω + i(ω − 1) = n → GFP (n) −i −i D ,D ,0 n 2 2 i odd i even o o ψ : ω ∈ H : iω + i(ω − 1) = n → GFP (n). −i −i D ,D ,−1 n 2 2 i even i odd In the following, we use the notation r to stand for a zigzag of length r on C ,ashiftof e e the points of C enclosed in the rectangle with opposite corners (1, −1) and (r, −2) and the notation r for a zigzag of length r on C , a shift of the points enclosed within the o o e e same rectangle on C . Given ω ∈ H , the map ψ stacks (ω − I{i even}) copies of the o −i zigzag i (whenever this is non-negative) vertically in increasing order and then removes a point from the bottom of each of the columns 1, 2, ... ,i, for each even i such that ω = 0 −i (giving the generalised Young diagram of an element of GFP (n)). Given ω ∈ H , D ,D ,0 2 2 the map ψ stacks (ω − I{i odd}) copies of the zigzag i (whenever this is non-negative) −i o vertically in increasing order and then removes a point from the bottom of each of the columns 1, 2, ... ,i, for each odd i such that ω = 0 (giving the generalised Young diagram −i of an element of GFP (n)). See Fig. 7 for examples of these maps. It is possible, but D ,D ,−1 2 2 tedious, to verify that these maps provide the necessary bijections. We leave this to the reader. Balázs et al. Res Math Sci (2022) 9:48 Page 21 of 46 48 ω =1 −7 n =19 ω =2 −5 m =4 ω =2 −4 −→ ω =1 −3 −11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 ω =(3, 0, 1, 2, 2, 0, 1, 1, 0, 1, 0, ...) ω =3 −1 ⏐ ⏐ e Since ω = 0, remove ⏐ −2 ⏐ 19,4 a point from the bottom of columns 1 and 2. n =19 m =4 Since ω = 0, remove −6 a point from the bottom of columns 1 through 6. ←− (a) A state ω ∈H and its image in GFP (19) D ,D ,0,4 2 2 ω =1 −8 n =19 ω =2 −6 m =4 ω =1 −4 −→ ω =2 −11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 −2 ω =(2, 2, 0, 1, 1, 2, 0, 1, 1, 0, 1, ...) ω =1 −1 ⏐ ⏐ Since ω = 0, remove −3 ⏐ ⏐ a point from the bottom 19,4 of columns 1,2 and 3. n =19 m =5 Since ω = 0, remove −7 a point from the bottom of columns 1 through 7. ←− − 3110 5 3110 (b) A state ω ∈H and its image in GFP (19) D ,D ,−1,5 2 2 e o Fig. 7 Bijections ψ and ψ 19,4 19,4 48 Page 22 of 46 Balázs et al. Res Math Sci (2022) 9:48 2,15 −→ 422 6441 0 0 220 −−−− 00 (a) An element of GFP (15) and its image in GFP (21) D ,D ,0 D ,D ,4 2 2 2 2 1,15 −→ − 422 644 0 0 2 200 −−− 00 (b) An element of GFP (15) and its image in GFP (19) D ,D ,−1 D ,D ,3 2 2 2 2 e o Fig. 8 Bijections φ and φ 2,15 1,15 We now consider the other values of the offset k. To tackle these cases it suffices, as in the Jacobi triple product case, to prove the following equivalences of generating functions: ⎨ (+1) f (˜q, t)˜q if k = 2, D ,D ,0 2 2 f (˜q, t) = D ,D ,k 2 2 2 ⎩ (+1) f (˜q, t)˜q if k = 2 + 1. D ,D ,−1 2 2 These follow naturally from a generalisation of Wright’s bijection φ , i.e. bijections φ :GFP (n) → GFP (n + ( + 1)), D ,D ,0 D ,D ,2 ,n 2 2 2 2 o 2 φ :GFP (n) → GFP (n + ( + 1) ) D ,D ,−1 D ,D ,2+1 ,n 2 2 2 2 defined for each n ≥ 0and  ∈ Z as follows. Given an element of GFP (n), the map D ,D ,0 2 2 |2|(|2|+1) φ adds the points inside the right angled triangle of size to either the left ,n 2 or top edge of its generalised Young diagram, depending on whether  ≥ 0or < 0, respectively. It then uses the new leading diagonal implied by the triangle to read off an element of GFP (n + ( + 1)). Given an element of GFP (n), the map φ D ,D ,2 D ,D ,−1 2 2 2 2 ,n |2+1|(|2+1|+1) adds the points inside the right-angled triangle of size to either the left or top edge of its generalised Young diagram, depending on whether  ≥ 0or < 0, respectively. It then uses the new leading diagonal implied by the triangle to read off an element of GFP (n + ( + 1) ). In both cases, the offset is supplied by the empty D ,D ,2+1 2 2 rows/columns coming from the triangle (in a similar fashion to Wright’s bijection). See e o Fig. 8 for examples of φ and φ .Notealsothatinbothcases thenumber m of distinct parts is clearly preserved and so the above maps restrict to give well defined bijections φ :GFP (n) → GFP (n + ( + 1)) D ,D ,0,m D ,D ,2,m ,n,m 2 2 2 2 o 2 φ :GFP (n) → GFP (n + ( + 1) ) D ,D ,−1,m D ,D ,2+1,m ,n,m 2 2 2 2 for each n, m ≥ 0and  ∈ Z. This proves the required equivalences of generating functions and hence completes the combinatorial justification of the identities in Theorem 1.1. Balázs et al. Res Math Sci (2022) 9:48 Page 23 of 46 48 4 Specialising to certain models In this section, we specialise the work of Sect. 3 to the ASEP(q, 1), 2-exclusion and 3- state models. For each of these models, the three variable identities in Theorem 1.1 will specialise to give two variable identities related to other known Jacobi style identities of combinatorial significance. 4.1 ASEP(q, 1) We will consider first the asymmetric simple exclusion process ASEP(q, 1) on ,as described by Redig et al. in [4]. This has asymmetry parameter 0 < q < 1and jump rates given by η −η −3 i i+1 p(η , η ) = q [η ] [2 − η ] and i i+1 i q i+1 q η −η +3 i i+1 q(η , η ) = q [2 − η ] [η ] . i i+1 i q i+1 q n −n q −q Here [n] := ,a q-deformation of n (we have that [n] → n as q → 1). q q −1 q−q The above rates might look strange at first sight, but they are carefully constructed in order to equip the corresponding process with “U (gl )-symmetry” (here U (gl )isthe q 2 q 2 quantum group associated with the Lie algebra gl ). The original motivation for construct- ing processes with “U (g)-symmetry” (for certain Lie algebras g) was the realisation that such processes have many explicit self-duality functions. The Markov generators for such processes are built by applying ground state transformations to quantum Hamiltonians associated with certain tensor products of representations of U (g). Roughly speaking the dimension of the representation relates to the number of particles allowed at a site, the dimension of g relates to the number of species of particle and tensors let us describe multi-site states. The Quantum Hamiltonian determines the jump rates. In the case of g = gl , the representations are the spin 2j representations for j ∈ Z (up to twist) and so onegetsafamilyASEP(q, j) of processes, each allowing up to 2j particles at a site. When j = , we recover classical ASEP and when j = 1 we get the process described above. We show that ASEP(q, 1) is a member of the blocking family. Conditions (B1) and (B2) clearly hold and the jump rates satisfy: (a) p(y, z) > q(z, y) for all y ∈{1, 2} and z ∈{0, 1}, p(1,0) p(2,1) −4 (b) = q = , q(0,1) q(1,2) 4 2 q [2] p(1,0)p(2,1)q(1,1)q(0,2) q −8 (c) = q · = 1. −4 2 q(0,1)q(1,2)p(2,0)p(1,1) q [2] Thus, (B3) is satisfied with the constants −2 −4 2 q [2] q q [2] 1 q q p = = and q = = asym asym −2 2 −4 −2 2 −4 [2] q + q 1 + q [2] q + q 1 + q q q and the functions y−z−2 −4 [z] q (1 + q )[3 − y] [3 − z] q q q f (z) = s(y, z) = . −4 [3 − z] q In this case, q˜ = q ,t = [2] and we have a one parameter family of product blocking measures of the form 0 η (2−η ) ∞ η (2−η ) i i −4η (i−c) i i 4(2−η )(i−c) i  i [2] q [2] q q q μ (η) = . −4(i−c) −8(i−c) 4(i−c) 8(i−c) (1 + [2] q + q ) (1 + [2] q + q ) q q i=−∞ i=1 Substituting the values of q˜ and t into Theorem 1.1 gives 4 4(+1) 2 4i 8i 2 2 S (q , [2] )q z = (1 + [2] q z + q z ) even q q ∈Z i≥1 48 Page 24 of 46 Balázs et al. Res Math Sci (2022) 9:48 4(i−1) −1 8(i−1) −2 ×(1 + [2] q z + q z ) 4i 8i 2 + (1 − [2] q z + q z ) i≥1 4(i−1) −1 8(i−1) −2 ×(1 − [2] q z + q z )) 4 4(+1) 2+1 4i 8i 2 2[2] S (q , [2] )q z = (1 + [2] q z + q z ) q odd q q i≥1 ∈Z 4(i−1) −1 8(i−1) −2 ×(1 + [2] q z + q z ) 4i 8i 2 − (1 − [2] q z + q z ) i≥1 4(i−1) −1 8(i−1) −2 ×(1 − [2] q z + q z )). −1 Since [2] = q +q , the quadratic terms on the RHS all factor and the products collapse as follows: 4i−1 4i+1 4i−5 −1 4i−3 −1 (1 + q z)(1 + q z)(1 + q z )(1 + q z ) i≥1 4i−1 4i+1 4i−5 −1 4i−3 −1 ± (1 − q z)(1 − q z)(1 − q z )(1 − q z ) i≥1 −1 −1 1 + q z 2i−1 2i−1 −1 = (1 + q z)(1 + q z ) 1 + qz i≥1 −1 −1 1 − q z 2i−1 2i−1 −1 ± (1 − q z)(1 − q z ) 1 − qz i≥1 ⎛ ⎞ 2i−1 2i−1 −1 2i−1 2i−1 −1 ⎝ ⎠ = (1 + q z)(1 + q z ) ∓ (1 − q z)(1 − q z ) . qz i≥1 i≥1 We have therefore proved the following identities. Theorem 4.1 4 (2+1) 2+1 2i−1 2i−1 −1 2 S (q , [2] )q z = (1 + q z)(1 + q z ) even q ∈Z i≥1 2i−1 2i−1 −1 − (1 − q z)(1 − q z ) i≥1 2 4 (2) 2 2i−1 2i−1 −1 2(1 + q ) S (q , [2] )q z = (1 + q z)(1 + q z ) odd q ∈Z i≥1 2i−1 2i−1 −1 + (1 − q z)(1 − q z ). i≥1 The RHS of these identities should look familiar. The first term is part of the product side of the Jacobi triple product, as seen in introduction (an alternative form is given in Sect. 3.3). Indeed, these two identities are isolating the odd/even terms, respectively (as mentioned in Sect. 3.3, this is the only reason for the sign changes on the RHS). Balázs et al. Res Math Sci (2022) 9:48 Page 25 of 46 48 We can now proceed in two ways. If we assume the Jacobi triple product, then Theorem 4.1 proves the following closed forms for the specialised normalising factors: 4 2 S (q , [2] ) =  (= f (q )), even q D ,D ,0 1 1 2j (1 − q ) j≥1 1 1 4 2 S (q , [2] ) =  = f (q ) . odd q D ,D ,0 1 1 2 2j 2 (1 + q ) (1 − q ) (1 + q ) j≥1 On the other hand, if we were able to find probabilistic explanations for these two equal- ities, then this would give a new probabilistic proof of the Jacobi triple product. However, the authors were unable to find such an explanation. To elaborate further, ASEP(q, 1) is a particular 0–1–2 system with blocking measure. However, the above equalities suggest that this blocking measure (and the one for its stood up process) should be intimately related to that of the 0-1 system ASEP(q, ) (i.e. classical ASEP) and its stood up process AZRP (which together prove the Jacobi triple product). Given the strange nature of the rates of ASEP(q, 1) it is not clear, at least to the authors, why this relationship should be expected. We can give explicit combinatorial explanations for these equalities. Let us look at the even case first. We have already seen that S (˜q, t) = f (˜q, t). Note that setting even D ,D ,0 2 2 4 −1 n m 4n −1 m 4n+m 4n+(m−2) q˜ = q and t = q + q sends q˜ t to q (q + q ) = q + q + ··· + 4n−(m−2) 4n−m q + q and so in order to prove the equality it suffices to show how m−1 elements of GFP (n) can be used to construct unique ordinary partitions of D ,D ,0,m 2 2 4n −(m −2i) into even parts for each 0 ≤ i ≤ m. Equivalently we must show how to obtain m m−2i unique ordinary partitions of 2n − . In order to do this, we take the generalised i 2 Young diagram corresponding to such a GFP, and for each square on the diagonal we colour in all intermediate white squares to the right and below. This almost creates a valid Young diagram but repeats in the rows of the GFP cause a problem, since the first entry in a repeat gives a row/column that is too short. To fix this, we add an extra black square to the end of such rows/columns, creating a valid Young diagram for a partition of 2n − . The other required partitions are obtained by adding more dots to this Young diagram. The only rows/columns that we are guaranteed to add dots to and still get a valid Young diagram are those corresponding to distinct entries in the rows of the GFP. For each of the choices of i such entries, we can add a dot onto the corresponding rows/columns, m−2i giving unique partitions of 2n − (for each 0 ≤ i ≤ m). See Fig. 9 for an example of the above construction (the colour grey is used to emphasise the squares that have been coloured black). The odd offset case is similar. We have already seen that tS (˜q, t) = f (˜q, t). Set- D ,D ,−1 odd 2 2 m m 4 −1 n m 4n+m 4n+(m−2) 4n−(m−2) ting q˜ = q and t = q +q sends q˜ t to q + q +···+ q + 1 m−1 4n−m 2 q as before. Multiplying by an extra q (to make the factor (1 + q ) on the LHS) gives 4n+(m+1) m 4n+(m−1) m 4n−(m−3) 4n−(m−1) q + q + ··· + q + q . In order to prove the 1 m−1 equality, it suffices to show how elements of GFP (n) can be used to construct D ,D ,0,−1 2 2 unique ordinary partitions of 4n − (m − 1 − 2i) into even parts for each 0 ≤ i ≤ m. Equiv- m m−1−2i alently we must show how to obtain unique ordinary partitions of 2n − .We i 2 do this in the same way as the even case, but being careful to also colour the intermediate squares in the first column and add a dot to the first column of the Young diagram if b is 1 48 Page 26 of 46 Balázs et al. Res Math Sci (2022) 9:48 −→ −→ (a) An element of GFP (14) and the associated partition of 2(14) − =27. D ,D ,0,2 2 2 {a } {b } {a ,b } 1 3 1 3 (b) Partitions of 28 and 29 coming from subsets of distinct entries of the GFP. Fig. 9 Partitions of 27, 28 and 29 coming from an element of GFP (14) D ,D ,0,2 2 2 −→ −→ − 422 3 200 3−1 (a) An element of GFP (16) and the associated partition of 2(16)− = 31. D ,D ,−1,3 2 2 {a } {b } {b } {a ,b } 1 1 2 1 1 {a ,b } {b ,b } {a ,b ,b } 1 2 1 2 1 1 2 Fig. 10 Partitions of 31, 32, 33 and 34 coming from an element of GFP (16) D ,D ,−1,3 2 2 not a distinct entry in row 2 of the GFP (the first column doesn’t correspond to a diagonal square). See Fig. 10 for an example. 4.2 Asymmetric particle-antiparticle exclusion process (3-state model) We now consider the asymmetric particle-antiparticle process with I ={−1, 0, 1},  = Z, asymmetry parameter 0 < q < 1, annihilation parameters a, a > 0, creation parameters γ , γ > 0 and jump rates as follows: • Particles (η = 1) jump left with rate q or right with rate 1 to empty sites, • Antiparticles (η =−1) jump left with rate 1 or right with rate q to empty sites, i Balázs et al. Res Math Sci (2022) 9:48 Page 27 of 46 48 Table 3 Jump rates p(η , η ) and q(η , η ), respectively, of i i+1 i i+1 the 3-state model η = 0 η = 1 η = 2 i+1 i+1 i+1 η =0 000 η =11 γ 0 η =2 210 η = 0 η = 1 η = 2 i+1 i+1 i+1 η =00 q 2q η =10 γqq η =2 000 • A particle–antiparticle neighbouring pair can be annihilated with rate a if η = 1and η =−1orrate a q if η =−1and η = 1, i+1 i i+1 • A particle-antiparticle pair can be created at two neighbouring empty sites with rate (i,i+1) (i,i+1) (i,i+1) (i,i+1) γ if η =−1and η = 1orrate γ q if η = 1and η =−1. i i+1 i i+1 Remark This process has been studied in the literature (see [2], for example), and it is known that the system can only have an i.i.d stationary distribution when the annihilation rate is twice the jump rate of a particle, i.e. when a = a = 2. We will consider this case from now on. We can view the 3-state model as a process on , with annihilation being a jump from a two particle site to an empty site and creation a jump from a one particle site to another one particle site. When thought of in this way, the jump rates are as given in Table 3. In order to be a member of the blocking family, we see that (B1) gives no constraints, (B2) is satisfied if and only if 0 <γ , γ ≤ 1, and the following must be satisfied: (a) p(y, z) > q(z, y) for all y ∈{1, 2} and z ∈{0, 1}, p(1,0) p(2,1) −1 (b) = q = , q(0,1) q(1,2) p(1,0)p(2,1)q(1,1)q(0,2) γ (c) 1 = = , i.e. 0 <γ = γ ≤ 1. q(0,1)q(1,2)p(2,0)p(1,1) Under this assumption, (B3) is then satisfied with the constants 1 q p = and q = asym asym 1 + q 1 + q and the functions (1 + q)if y = z = 1, ⎪ γ ⎧ ⎪ ⎪ ⎪ 0if z = 0, 2γ (1 + q)if y = 1and z = 2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨  ⎨ if z = 1, f (z):= s(y, z):= 2γ (1 + q)if y = 2and z = 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 ⎪ γ if z = 2, ⎪ (1 + q)if y = z = 2, γ ⎪ 2 0if y = 3or z = 3. 48 Page 28 of 46 Balázs et al. Res Math Sci (2022) 9:48 In this case, q˜ = q and t = and we have a one-parameter family of blocking measures of the form 1   1 I{η =1} I{η =1} i i 2 2 2 −(i−c)η 2 (2−η )(i−c) 0 i ∞ i q q γ γ μ (η) =   . 2 2 −(i−c) −2(i−c) (i−c) 2(i−c) 1 + q + q 1 + q + q i=−∞ i=1 γ γ Since γ can be almost freely chosen (there is only the constraint that 0 <γ ≤ 1), applying Theorem 1.1 to this subfamily will not tell us anything new (we will get the same three variable identity but with domain restricted to t ≥ 2). However, certain members of this subfamily can give interesting two variable identities. As an example, we will choose the creation parameter to be γ = (the inverse of the annihilation parameter a = 2). For this process, we then have q˜ = q and t = 2. Substituting these values of q˜ and t into Theorem 1.1 gives (+1) 2 i 2i 2 i−1 −1 2(i−1) −2 2 S (q, 2)q z = (1 + 2q z + q z )(1 + 2q z + q z ) even ∈Z i≥1 i 2i 2 i−1 −1 2(i−1) −2 + (1 − 2q z + q z )(1 − 2q z + q z )), i≥1 (+1) 2+1 i 2i 2 i−1 −1 2(i−1) −2 4 S (q, 2)q z = (1 + 2q z + q z )(1 + 2q z + q z ) odd ∈Z i≥1 i 2i 2 i−1 −1 2(i−1) −2 − (1 − 2q z + q z )(1 − 2q z + q z )). i≥1 Once again the quadratic terms on the RHS all factor and so we have proved the following identities Theorem 4.2 (+1) 2 i i−1 −1 2 2 S (q, 2)q z = ((1 + q z)(1 + q z )) even i≥1 ∈Z i i−1 −1 2 + ((1 − q z)(1 − q z )) i≥1 (+1) 2+1 i i−1 −1 2 4 S (q, 2)q z = ((1 + q z)(1 + q z )) odd ∈Z i≥1 i i−1 −1 2 − ((1 − q z)(1 − q z )) . i≥1 Again the first term on the RHS looks familiar and is part of the square of the product side of the Jacobi triple product. Indeed, if we assume Jacobi triple product, then we find that 1 k(k+1) i i−1 −1 2 k ((1 + q z)(1 + q z )) =  q z (1 − q ) i≥1 i≥1 k∈Z k (k +1) (k−k )(k−k +1) + k 2 2 = q z . i 2 (1 − q ) i≥1 k∈Z k ∈Z Balázs et al. Res Math Sci (2022) 9:48 Page 29 of 46 48 The term corresponding to k = 2 can be written as (using Jacobi triple product in the second equality, written as in the introduction but setting z = 1) k (k +1) (2−k )(2−k +1) + 2 2 2 q z i 2 (1 − q ) i≥1 k ∈Z 1  2 (k −) (+1) 2 =  q q z i 2 (1 − q ) i≥1 k ∈Z ⎛ ⎞ 2i−1 2 2i (1 + q ) (1 − q ) (+1) 2 ⎝ ⎠ = q z i 2 (1 − q ) i≥1 ⎛ ⎞ 2i−1 2 i (1 + q ) (1 + q ) (+1) 2 ⎝ ⎠ = q z . (1 − q ) i≥1 Similarly the term corresponding to k = 2 + 1 can be written as (using Jacobi triple product in the second equality, written as in Sect. 3.3 but setting z = 1) k (k +1) (2−k +1)(2−k +2) + 2+1 2 2 q z i 2 (1 − q ) i≥1 k ∈Z 1   2 (k −)(k −−1) (+1) 2+1 =  q q z i 2 (1 − q ) i≥1 k ∈Z ⎛ ⎞ 2i 2 2i (1 + q ) (1 − q ) 2 (+1) 2+1 ⎝ ⎠ = 2 q z i 2 (1 − q ) i≥1 ⎛ ⎞ 2i 2 i (1 + q ) (1 + q ) 2 (+1) 2+1 ⎝ ⎠ = 2 q z . (1 − q ) i≥1 Thus, we see that, assuming Jacobi triple product, the identities are equivalent to the equalities 2i−1 2 i (1 + q ) (1 + q ) S (q, 2) = , even (1 − q ) i≥1 2i 2 i (1 + q ) (1 + q ) S (q, 2) = . odd (1 − q ) i≥1 As in the previous section, it would be interesting to find purely probabilistic proofs of these product forms. The more complicated nature of the products suggests that this is non-trivial. Alternatively one could have not assumed Jacobi triple product in the above, but instead assumed Theorem 4.1. The identities would then give (non-trivial) relations between the 4 4 four specialised normalising factors S (q, 2),S (q, 2),S (q , [2] )and S (q , [2] ) even odd even q odd q that we would have proved probabilistically without additional assumptions. While the identities in Theorem 4.2 might seem unnatural, they do have a combinatorial i 2 interpretation in terms of 2-coloured GFP’s. The product (1 + q ) is the generating i≥1 function for coloured partitions of n into red/blue parts where each red/blue part appears 48 Page 30 of 46 Balázs et al. Res Math Sci (2022) 9:48 at most once (for example we allow 5 = 2 + 2 + 1but not5 = 2 + 2 + 1). The pair of 2-coloured partitions 2 +2 +1and 2 +2 +1 are counted as the same in the above product, and so to avoid overcounting we favour a particular colour when listing repeats. By the “General Principle” of Andrews, we have that: i i−1 −1 2 k ((1 + q z)(1 + q z )) = f (q)z , C ,C ,k 2 2 i≥1 k∈Z where f (q) is the generating function for the sets GFP (n)ofGFP’s of n with C ,C ,k C ,C ,k 2 2 2 2 offset k and each row being a 2-coloured partition. The content of the above identities is then that (+1) S (q, 2)q if k = 2, even f (q) = C ,C ,k 2 2 2 (+1) 2S (q, 2)q if k = 2 + 1. odd Thus, the two specialised normalising factors satisfy f (q) = 2S (q, 2) and C ,C ,−1 2 2 odd f (q) = S (q, 2) and so have a natural combinatorial interpretation. These two base C ,C ,0 even 2 2 cases are both explicitly clear since we know from Sect. 3.3 that S (˜q, t) = f (˜q, t) even D ,D ,0 2 2 and tS (˜q, t) = f (˜q, t), and each element of GFP (n)and GFP (n) D ,D ,−1 D ,D ,0,m D ,D ,−1,m odd 2 2 2 2 2 2 can be 2-coloured in 2 ways (exactly what is counted when setting t = 2). The other offset cases can be proved in the usual fashion, by the equalities of generating functions: ⎨ (+1) f (q)q if k = 2, C ,C ,0 2 2 f (q) = C ,C ,k 2 2 (+1) f (q)q if k = 2 + 1. C ,C ,−1 2 2 e o (The proof follows from the maps φ , φ of Sect. 3.3, but by using coloured generalised ,n ,n Young diagrams). The function f (q) is the function C (q) defined by Andrews on p.7 of [1]. Indeed, C ,C ,0 2 2 2 the specialised normalising factor is: 2 3 4 5 6 7 8 S (q, 2) =1+4q +9q +20q + 42q + 80q + 147q + 260q + 445q + ··· even which agrees with the expansion of C (q) found on p.8 of the same book. In Corollary 5.2 of this book, Andrews uses Jacobi triple product to prove a product formula for C (q), which is equivalent to the product formula we found above for S (q, 2). Andrews’ book even only considers GFP’s with offset 0 and so does not give a similar analysis of the function f (q). However, using MAGMA we were able to compute that C ,C ,−1 2 2 2 3 4 5 6 7 8 f (q) =2+4q +12q + 24q + 50q + 92q + 172q + 296q + 510q + ··· C ,C ,−1 2 2 which agrees with 2S (q, 2), as expected, and appears to have the same coefficients as odd twice OEIS sequence A137829 [7], implying the product form we derived above. The function C (q)isone of afamilyoffunctions C (q), counting GFP’s of offset 0 with rows having a similar condition to the above but with k colours. The whole family of functions is studied in Andrews’ book. In general they are not given by products but can be shown to be sums of products. It would be interesting to know whether there exists a k- state model for each k, whose stood up process and stationary blocking measures provide normalising factors relating to these functions (and their corresponding non-zero offset analogues). 4.3 Asymmetric 2-exclusion Now we consider the asymmetric 2-exclusion process on  with asymmetry parameter 0 < q < 1. The nonzero left/right jump rates are Balázs et al. Res Math Sci (2022) 9:48 Page 31 of 46 48 p(η , η ) = I{η = 0}I{η = 2} and q(η , η ) = qI{η = 0}I{η = 2}. i i+1 i i+1 i i+1 i+1 i We show that this is a member of the blocking family. Conditions (B1) and (B2) clearly hold and the jump rates satisfy: (a) p(y, z) > q(z, y) for all y ∈{1, 2} and z ∈{0, 1}, p(1,0) p(2,1) −1 (b) = q = , q(0,1) q(1,2) p(1,0)p(2,1)q(1,1)q(0,2) (c) = 1. q(0,1)q(1,2)p(2,0)p(1,1) Thus, (B3) is satisfied with the constants 1 q p = and q = , asym asym 1 + q 1 + q and the functions, (1 + q) for y, z ∈{1, 2}, f (z) = I{z = 0},s(y, z) = 0if y = 3or z = 3. In this case, q˜ = q, t = 1 and we have a one-parameter family of product stationary blocking measures of the form −η (i−c) (2−η )(i−c) i  i q q μ (η) = . −(i−c) −2(i−c) (i−c) 2(i−c) (1 + q + q ) (1 + q + q ) i≤0 i≥1 Substituting the values of q˜ and t into Theorem 1.1 gives Theorem 4.3 (+1) 2 i 2i 2 i−1 −1 2(i−1) −2 2 S (q, 1)q z = 1 + q z + q z 1 + q z + q z even ∈Z i≥1 i 2i 2 i−1 −1 2(i−1) −2 + 1 − q z + q z 1 − q z + q z i≥1 (+1) 2+1 i 2i 2 i−1 −1 2(i−1) −2 2 S (q, 1)q z = 1 + q z + q z 1 + q z + q z odd ∈Z i=1 i 2i 2 i−1 −1 2(i−1) −2 − 1 − q z + q z 1 − q z + q z . i=1 It is clear that this specialisation has a natural combinatorial meaning. Recall that in Sect. 3.3 we used the “General Principle” to expand i 2 2i −1 i−1 −2 2(i−1) k (1 + tzq˜ + z q˜ )(1 + tz q˜ + z q˜ ) = f (˜q, t)z D ,D ,k 2 2 i≥1 k∈Z with f (˜q, t) being the two variable generating function for the sets GFP (n) D ,D ,k D ,D ,k,m 2 2 2 2 defined earlier. Setting t = 1 is naturally interpreted as not distinguishing GFP’s by their number of distinct parts per row, i.e. f (˜q, 1) = f (˜q), the one variable generating D ,D ,k D ,D ,k 2 2 2 2 function for the sets GFP (n). The content of the above identities is then that D ,D ,k 2 2 ⎨ (+1) S (q, 1)q if k = 2, even f (q) = D ,D ,k 2 2 2 (+1) S (q, 1)q if k = 2 + 1. odd 48 Page 32 of 46 Balázs et al. Res Math Sci (2022) 9:48 e o e o The maps ψ , ψ , φ and φ of Sect. 3.3 give explicit proofs for all of these equalities, as n n ,n ,n expected. Let’s consider the two base cases in more detail. The function f (q) is the function D ,D ,0 2 2 (q) defined by Andrews on p.6 of [1]. The specialised even normalising factor is: 2 3 4 5 6 7 8 S (q, 1) = 1 + q + 3q + 5q + 9q + 14q + 24q + 35q + 55q + ··· even which agrees with the expansion of  (q) on p.7 of the same book, as expected. An interesting result in this book is Corollary 5.1, which uses the Jacobi triple product and other results to prove that (q) =  . i 12i−10 12i−9 12i−3 12i−2 (1 − q )(1 − q )(1 − q )(1 − q )(1 − q ) i≥1 So the specialised even normalising factor can be expressed as the above product. Andrews’ book only considers GFP’s with offset 0 and so does not give a similar analysis of the function f (q). However, using MAGMA we were able to compute that D ,D ,−1 2 2 2 3 4 5 6 7 8 f (q) = 1 + 2q + 3q + 6q + 10q + 16q + 26q + 40q + 60q + ··· D ,D ,−1 2 2 which agrees with S (q, 1) and appears to have the same coefficients as OEIS sequence odd A201077 [8], implying that S (q, 1) = f (q) odd D ,D ,−1 2 2 = . 2i−1 2 12i−8 12i−6 12i−4 12i (1−q ) (1−q )(1−q )(1−q )(1−q ) i≥1 This discussion raises two interesting questions. Firstly, is there a probabilistic expla- nation for the above product forms? These products are quite complicated and it is not clear a priori that we should expect such a factorisation. It could be possible that there is an alternative way to stand up the 2-exclusion process, giving a more natural normalising factor. Secondly, in this case, the ASEP(q, 1) case and the 3-state model case the specialised normalising factors appear to be products. Could it be that the unspecialised normalising factors S (˜q, t)and S (˜q, t) are products? even odd 5 Asymmetric k-exclusion It is natural to ask whether we can generalise the results of this paper to higher order particle systems with blocking measure. Unfortunately, as remarked in Sect. 3.2 the “stood up” process in general is not guaranteed to have a product stationary blocking measure. However, we will now see that the asymmetric k-exclusion processes for k ≥ 1are sufficiently well behaved and lead to an interesting family of combinatorial identities, generalising the ones found in the 2-exclusion section. The asymmetric k-exclusion process is the particle system with I ={0, 1, ... ,k},  = Z, asymmetry parameter 0 < q < 1 and jump rates p(η , η ) = I{η = 0}I{η = k} and q(η , η ) = qI{η = 0}I{η = k}. i i+1 i i+1 i i+1 i+1 i We now show that this is member of the blocking family. Conditions (B1) and (B2) clearly hold so it suffices to check (B3). We see that the jump rates are described by the constants 1 q p = q = asym asym 1 + q 1 + q and functions (1 + q) for y, z ∈{1, 2, ... ,k}, f (z) = I{z = 0} and s(y, z) = 0if y = k + 1or z = k + 1. Balázs et al. Res Math Sci (2022) 9:48 Page 33 of 46 48 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 0 −1 (a) Ground state η (b) Ground state η −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −2 −3 (c) Ground state η (d) Ground state η Fig. 11 Four ground states of 4-exclusion Thus, k-exclusion is a member of the blocking family and by Theorem 2.1 has one param- eter family of product stationary blocking measures 0 ∞ −(i−c)η (k−η )(i−c) i  i q q μ (η) = , c c k(i−c) Z (q) q Z (q) i i i=−∞ i=1 c −(i−c)y where Z (q) = q is the normalising factor. y=0 By definition of the state space, every η ∈  hasaleftmostparticleand aright most hole, i.e. η = 0 for small enough i and η = k for big enough i. By asymmetry, it then i i −m follows that the ground states of  are all shifts of the following k ground states η , for m ∈{0, 1, ... ,k − 1} with 0if i < 0, −m η = m if i = 0, k if i > 0. See Fig. 11 for example when k = 4. 5.1 Ergodic decomposition of ∞ 0 The quantity N(η):= (k − η ) − η is finite and is conserved by the dynamics i i i=1 i=−∞ of the system. Thus, just as in Sect. 3.1 we can decompose  =  , into irreducible n∈Z components  :={η ∈  : N(η) = n}. Note now that the left shift operator τ gives a n n−k n bijection  − →  (i.e. if η ∈  then, N(τη) = n − k). −m −m Remark Since N(η ) =−m for each m ∈{0, 1, ... ,k − 1},the shifts of η have con- served quantity in kZ−m and give the ground states for the (−m mod k)part n∈kZ−m of . n,c c n We now calculate ν (·):= μ (·|N(·) = n), the unique stationary distribution on  . Lemma 5.1 The following relation holds c kc−N(η) c μ (τη) = q μ (η). This gives the recursion c n−kc c μ ({N = n}) = q μ ({N = n − k}). 48 Page 34 of 46 Balázs et al. Res Math Sci (2022) 9:48 The proof is similar to that of Lemma 3.1 (both claims are special cases of Lemma 6.1 and Corollary 6.2 in [3]). The general solution of this recursion is (n+m)(n+k−m) c −(n+m)c c 2k μ ({N = n}) = q μ ({N =−m}) if n ∈ kZ − m with m ∈{0, 1, ... ,k − 1}. Since there is a dependence on the class of n modulo k we will need to calculate the probabilities μ ({N(η) ≡−m mod k}) for m ∈{0, 1, ... ,k − 1} in order to finish our n,c calculation of ν . Lemma 5.2 ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ k−1 ∞ k−1 rj c −rm c ⎝ ⎝ ⎝ ⎠ ⎠ ⎠ μ ({N ≡−m mod k}) = 1 + ζ 1 + (ζ − 1)μ (j) , k k r=1 i=−∞ j=1 th where ζ is a primitive k root of unity. Proof Define the partial conserved quantity a 0 N (η) = (k − η ) − η for a ≥ 1 a i i i=1 i=−a and note that N (η) → N(η)as a →∞. Consider the character group Z/kZ := Hom Z/kZ, C ={χ , χ , ... , χ }, 0 1 k−1 where χ (1) = ζ for i ∈{0, 1, ... ,k − 1} and ζ aprimitive k-th root of unity. Note that i k χ = χ . For each a ≥ 1 we define the random variables a a Y := χ (−N (η)) = χ η = χ (η ). a 1 a 1 i 1 i i=−a i=−a Since η → 0as i →−∞ and η → k as i →∞ we have that Y → Y as a →∞, where i i a ∞ ∞ Y := χ (−N(η)) = χ ( η ) = χ (η ). 1 1 i 1 i i=−∞ i=−∞ c r We now compute the moments E [Y ] for r ∈{0, 1, ... ,k − 1} (the expectation being with respect to μ ). Since |Y |= 1 for all a ≥ 1, dominated convergence applies and by the product structure of μ we have that ∞ k ∞ k c r c r r c c E [Y ] = lim E [Y ] = χ (j) μ (j) = χ (j)μ (j) . 1 r a i i a→∞ i=−∞ j=0 i=−∞ j=0 On the other hand, we have that k−1 c r r c E [Y ] = χ (m) μ − N(η) ≡ m mod k m=0 k−1 = χ (m)μ N(η) ≡−m mod k . m=0 Balázs et al. Res Math Sci (2022) 9:48 Page 35 of 46 48 Hence, we have the linear system of equations ∞ k k−1 c c χ (j)μ (j) = χ (m)μ N(η) ≡−m mod k . r r m=0 i=−∞ j=0 By orthogonality of characters, we get k−1 ∞ k c c μ (N(η) ≡−m mod k) = χ (m) χ (j)μ (j) r r r=0 i=−∞ j=0 k−1 ∞ k = χ (−m) χ (j)μ (j) r r r=0 i=−∞ j=0 ⎛ ⎛ ⎞ ⎞ k−1 ∞ k rj −rm c ⎝ ⎝ ⎠ ⎠ = ζ ζ μ (j) k k r=0 i=−∞ j=0 ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ k−1 ∞ k−1 rj −rm ⎝ ⎝ ⎝ ⎠ ⎠ ⎠ = 1 + ζ 1 + (ζ − 1)μ (j) . k k i r=1 i=−∞ j=1 Remark A priori this appears to be complex-valued. However, it is in fact real valued since it is fixed by complex conjugation. When k = 2 this is clearly real valued and agrees with Lemma 3.2 (recalling that t = 1 for 2-exclusion). If we combine Lemma 5.1 and Lemma 5.2, we find the form of the unique stationary n,c n distribution ν on  . Proposition 5.3 For n ∈ kZ − mwithm ∈{0, 1, ... ,k − 1}, the stationary distribution on  is given by 0 ∞ k(+1)  −(i−c)η  (k−η )(i−c) q i q i −m−kc k q I{N(η) = n} k(i−c) Z (q) q Z (q) ∈Z i=−∞ i=1 n,c ν (η) =    . (n+m)(n+k−m) rj −(n+m)c k−1 −rm ∞ k−1 c 2k q 1 + ζ 1 + (ζ − 1)μ (j) r=1 i=−∞ j=1 i k k Remark As in Sect. 3.1, these distributions are independent of c but we will need to stress the dependence of both the numerator and denominator on c. k rj −j(i−c) ζ q rj k−1 j=0 c k Notice that 1 + (ζ − 1)μ (j) = for each r ∈{1, ... ,k − 1}. Defining j=1 k i Z (q) rj c −j(i−c) n,c W (q):= ζ q for each such r we can write ν as i,r j=0 n,c ν (η) 0 ∞ k(+1) −m−kc −(i−c)η (k−η )(i−c) i i k q q q I{N(η) = n} ∈Z i=−∞ i=1 =    . (n+m)(n+k−m) −(n+m)c k−1 −rm k(i−c) c c k(i−c) c c 2k q q Z (q)Z (q) + ζ q W (q)W (q) i≥1 −i+1 i r=1 i≥1 −i+1,r i,r when n ∈ kZ − m with m ∈{0, 1, ... ,k − 1}. c c Remark When k = 2, there is only the possibility r = 1and W (q) = W (q, 1), as i,1 i expected (here W (q, t) is the function defined in Sect. 3.1). The corresponding stationary distributions then agree with the ones for 2-exclusion given by Proposition 3.3 (after setting t = 1). 48 Page 36 of 46 Balázs et al. Res Math Sci (2022) 9:48 10 14 3 9 13 2 6 8 12 1 4 5 7 11 LMP RMH -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 (A) A 4-exclusion state η. (B) Its stood up state ω. Fig. 12 Standing up map on 4-exclusion −11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 −11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 (a) ω (b) ω −11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 −11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 −2 −3 (c) ω (d) ω Fig. 13 Four ground states of stood up 4-exclusion 5.2 Standing up/laying down In this section, we transfer the dynamics on  to that of a restricted particle system on <0 Z by using the same “standing up” method as described in Sect. 3.2. By doing this we ≥0 obtain an alternative characterisation of the stationary distributions given in Proposition 5.3. n th We recall the standing up map. Given η ∈  let S being the site of the r particle when reading left to right, bottom to top. The corresponding stood up state is then n <0 T (η) = ω ∈ Z ,with ω = S − S .See Fig. 12 for an example with k = 4. −r r+1 r ≥0 n <0 A priori the “standing up” map T is an injection into Z . However, since η ≤ k for ≥0 all i, the image of T lies in the restricted state space <0 H :={ω ∈ Z : ω = ω = ··· = ω = 0 ⇒ ω = 0, ∀i > 0}. −i −i−1 −i−(k−2) −i−(k−1) ≥0 Since η = k for i large ω must coincide far to the left with one of the following states −m −m ω , for m ∈{0, 1, ... ,k − 1},definedby ω = I{i ∈ kZ + m} (Fig. 13). More precisely, −i m is uniquely determined by n ∈ kZ − m. −m −m −m Remark Note that all shifts of the ground state η ∈  stand up to give ω for each m ∈{0, 1, ... ,k − 1}. This is in direct analogy with the 2-exclusion case. k−1 n −m −m We now see that the image of T lies in H := H , where the disjoint sets H are m=0 defined as −m −m H :={ω ∈ H : ∃N >0s.t ω = ω ∀i ≥ N }. −i −i −m Given ω ∈ H we let D (ω) be the minimum such N in the above. −m n n −m Lemma 5.4 For n ∈ kZ − mwithm ∈{0, 1, ... ,k − 1},wehaveT ( ) = H . Proof It suffices to show surjectivity of T for each n. Balázs et al. Res Math Sci (2022) 9:48 Page 37 of 46 48 Table 4 Jump rates p (ω , ω ) and q (ω , ω ), ω −r −r+1 ω −r −r+1 respectively ω ≥ 0 −r+1 ω =00 −r k−1 ω =11 − I{ω = 0} −r −r−j j=1 ω ≥21 −r ω = 0 ω = 1 ω ≥ 2 −r+1 −r+1 −r+1 k−1 ω ≥00 q(1 − I{ω = 0}) q −r −r+1+j j=1 Table 5 Boundary jump rates for the stood up process, ω Rate into the boundary Rate out of the boundary ω =00 q −1 k−1 ω =11 − I{ω = 0} −1 −r−j j=1 ω ≥21 −1 −m n Take n ∈ kZ − m and ω ∈ H , then construct the state η ∈  having LMP at site D (ω) −m n + D (ω) − I{D (ω) ∈ / kZ − m} −m −m S = + 1 − ω 1 −i i=1 th n n and r particle at site S = S + ω .Itisclear that T (η) = ω and hence T is r r−1 1−r surjective. Just as before we call these inverse maps the “laying down“ maps. Using the “standing up” maps we define a particle system on H whose dynamics are inherited from those on . In particular right jumps in η correspond to right jumps in ω and similarly for left jumps. The explicit right/left jump rates are given in Table 4 for r ≥ 2. Since the “stood up ” process is only defined on the negative half integer line, we must consider what happens at the boundary site. As in Sect. 3.2, we will consider an open infinite type boundary (Table 5). To find the stationary distribution for the “stood up” process, we first consider the ∗ <0 unrestricted process ω ∈ Z , i.e. the process described by the same jump rates as ω but ≥0 where the number of consecutive zeros is not restricted. It is clear that the unrestricted process is simply the zero-range process which is a member of the blocking family [3], with one parameter family of blocking measures given by the marginals ∗,cˆ (i+cˆ)z (i+cˆ) π (z) = q (1 − q ). −i By considering dynamics at the boundary, we can fix the value of cˆ and so have a sin- ∗,cˆ gle product blocking measure. We suppose that π satisfies detailed balance over this boundary edge, i.e. ∗,cˆ ∗,cˆ π (y) · “rate into the boundary” = π (y − 1) · “ rate out of the boundary” for all y ≥ 1. −1 −1 Thus, for all y ≥ 1wehave (1+cˆ)y (i−c) (1+cˆ)(y−1) (i−c) q (1 − q ) = q (1 − q )q 48 Page 38 of 46 Balázs et al. Res Math Sci (2022) 9:48 and hence cˆ = 0, giving the stationary blocking measure ∗ ∗ iω i −i π (ω ) = q (1 − q ). i≥1 Now that we have the stationary distribution for the unrestricted process we consider k−1 −m the restriction to H and find the stationary measure. Recall that H = H ,and m=0 −m −m note that for each H is the irreducible component of the ground state ω .Wedefine −m ∗ stationary measures π on these irreducible components in terms of π . It seems natural −m ∗ −m ∗ to define these measures as π (·) = π (·|· ∈ H ). However, w.r.t π the probability of being in any irreducible component is zero and so these quantities are undefined. To rectify this, we use a similar formal reasoning as we did in Sect. 3.2 in order to give the stationary distributions in the following form: −m −m φ (ω )I{ω ∈ H } −i −i i≥1 −m π (ω) = , −m φ (ω ) −i −i −m i≥1 ω ∈H where ∗ ⎨ i(ω −1) −i π (ω ) q if i ∈ kZ + m, −i −m −i φ (ω ) = = −i −i −m iω π (ω ) −i q otherwise. −i −i These are given explicitly in the following proposition. −m Proposition 5.5 For each m ∈{0, 1, ... ,k − 1} the unique stationary measure on H is iω + i(ω −1) −i −i i∈ /kZ+m i∈kZ+m −m π (ω) = , (k) S (q) −m iω + i(ω −1) −i −i (k) i∈ /kZ+m i∈kZ+m where S (q) = q is the normalising factor with respect to −m −m ω ∈H −m H . −m Proof The result once again follows from Proposition 5.10 of [6] since each π is a restriction of π . 5.3 Identities n n By Lemma 5.4, the standing up transformation T describes a bijection between −m n and H , for the unique value m ∈{0, 1, ... ,k − 1} such that n ∈ kZ − m.Since T preserves the dynamics of the corresponding processes, we get an equality of measures, n,c −m n ν (η) = π (T (η)) for this value of m. −m −m Recall that k-exclusion has k ground states up to shift; η ∈  for m ∈{0, 1, ... ,k − −m −m −m −m,c −m −m −m 1}, satisfying T (η ) = ω .Thus, ν (η ) = π (ω ) and so by Proposition 5.3 and Proposition 5.5 we have the following identities for m ∈{0, 1, ... ,k − 1} (after rearrangement): k(+1) (k) −m−(k−m)c k(i−c) c c k S (q)q = q Z (q)Z (q) −m −i+1 i ∈Z i≥1 Balázs et al. Res Math Sci (2022) 9:48 Page 39 of 46 48 ⎛ ⎞ k−1 −rm k(i−c) c c ⎝ ⎠ + ζ q W (q)W (q) . −i+1,r i,r r=1 i≥1 c c −c Writing Z (q)and W (q) explicitly and letting z = q proves the following identities. i i,r Theorem 1.3 For 0 < q < 1,z = 0 and m ∈{0, 1, ... ,k − 1} k(+1) (k) −m k−m k S (q)q z −m ∈Z ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ k−1 k k −rm −αr αi α αr α(i−1) −α ⎝ ⎝ ⎠ ⎝ ⎠ ⎠ = ζ ζ q z ζ q z . k k k r=0 i≥1 α=0 α=0 Remark When k = 2, these agree with the two identities coming from 2-exclusion in Theorem 4.3. We now discuss the combinatorial significance of these identities. Note first that i 2i k (1 + q + q + ··· q ) is the generating function for ordinary partitions of n with i≥1 each part appearing at most k times. By the “General Principle” of Andrews, we can then interpret the first term on the RHS as: i 2i 2 ki k i−1 −1 2(i−1) −2 k(i−1) −k (1 + q z + q z + ··· + q z )(1 + q z + q z + ··· + q z ) i≥1 = f (q)z D ,D ,k k k k ∈Z where f (q) is the generating function for GFP (n). The function f (q)is D ,D ,k D ,D ,k D ,D ,0 k k k k k k the function  (q)definedbyAndrews on p.6of[1] (the special case k = 2 appeared earlier when considering 2-exclusion). The case of nonzero offset is not considered in the book. The other terms on the RHS might look unwieldy at first glance but are in fact the above −r but with change of variables z → ζ z. We saw this in all previous examples; ζ =−1 and this was providing the sign changes in the identities, the purpose of which were to isolate odd/even terms of another identity. Here we have the same behaviour, but now we are using kth roots of unity to isolate the z terms where k lies in a fixed class mod k.To justify this we expand the whole RHS; for m ∈{0, 1, ... ,k − 1},wehave ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ k−1 k k −rm −αr αi α αr α(i−1) −α ⎝ ⎝ ⎠ ⎝ ⎠ ⎠ ζ ζ q z ζ q z k k r=0 i≥1 α=0 α=0 k−1 −rm −rk k = ζ ζ f (q)z D ,D ,k k k k k r=0 k ∈Z ⎛ ⎞ k−1 −r(m+k ) ⎝ ⎠ = ζ f (q)z D ,D ,k k k k k ∈Z r=0 = k f (q)z . D ,D ,k k k k ≡−m mod k Given the above, we now see that our identities are equivalent to the equalities k(+1) (k) −m f (q) = S (q)q if k = k − m with m ∈{0, 1, ... k − 1}. D ,D ,k −m k k 48 Page 40 of 46 Balázs et al. Res Math Sci (2022) 9:48 ω−8 =1 n =29 ω =2 −7 −→ −12−11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 ω =1 −5 ω =(3, 2, 0, 1, 1, 0, 2, 1, 1, 0, 0, 1, ...) ω =1 −4 ω =2 −2 (0) ψ ⏐ ω−1 =3 ⏐ Since ω =0, remove a −3 point from the bottom of columns 1,2 and 3. ←− Since ω = 0,removea −6 n =29 point from the bottom of columns 1 through 6. (a) A state ω ∈H and its image in GFP (29) D ,D ,0 3 3 ω =1 −8 n =25 ω =1 −6 −→ ω =1 −5 −13−12−11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 ω−4 =2 ω =(0, 2, 2, 2, 1, 1, 0, 1, 0, 1, 0, 0, 1...) ω =2 −3 ω =2 −2 (1) ψ ⏐ ⏐ ⏐ Since ω =0, remove −1 a point from the bottom of column 1. n =25 ←− Since ω =0, remove a −7 point from the bottom of columns 1 through 7. − 2100 66 4 2 0 −1 (b) A state ω ∈H and its image in GFP (25) D3,D3,−1 (0) (1) (2) Fig. 14 Bijections ψ , ψ and ψ (cont.) 29 25 19 Balázs et al. Res Math Sci (2022) 9:48 Page 41 of 46 48 ω =1 −7 n =19 ω =1 −6 −→ ω =1 −4 −12−11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 ω =(3, 0, 2, 1, 0, 1, 1, 1, 0, 0, 1, 0, ...) ω =2 −3 ω =3 ⏐ −1 (2) ψ ⏐ ⏐ Since ω = 0, remove a −2 point from the bottom of columns 1 and 2. n =19 ←− Since ω = 0, remove a −5 point from the bottom of columns 1 through 5. ω =0 −5 −− 1100 6 3 3100 −2 (c) A state ω ∈H and its image in GFP (19) D ,D ,−2 3 3 Fig. 14 continued Once again, we have proved this probabilistically but are also interested in explicit combi- natorial explanations. In a similar vein to 2-exclusion, we find that there are k base cases, (k) f (q) = S (q) for m ∈{0, 1, ... ,k −1}. These can be proved by adapting the maps D ,D ,−m −m k k e o ψ , ψ of Sect. 3.3 to give maps: n n ⎧ ⎫ ⎨ ⎬ (m) −m ψ : ω ∈ H : iω + i(ω − 1) = n → GFP (n), −i −i D ,D ,−m n k k ⎩ ⎭ i∈ /kZ+m i∈kZ+m for n ≥ 0and m ∈{0, 1, ... ,k − 1}. We must first describe the process of assigning generalised Young diagrams to GFP’s with the k-repetition condition. An element of GFP (n)with m ∈{0, 1, ... ,k − 1} D ,D ,−m k k can be assigned a generalised Young diagram on the set of points C ={(n ,n ) ∈ m 1 2 Z | n + n ≡ m mod k} as follows. The subset corresponding to such an element with 1 2 s entries on the top row consists of the s leading diagonal points (m + 1, −1), (m + 1 1 2, −2), ... , (m + s , −s ), the first a points of C to the right of (m + i, −i) and the first b 1 1 i m i points of C under (i, m −i). Note that the points (i, m −i) for 1 ≤ i ≤ m are not included. (m) e o The maps ψ arethendefinedinasimilarfashion to ψ , ψ . In the following, we use n n the notation r to stand for a wave of length r on C , a shift of the points of C enclosed (m) m m 48 Page 42 of 46 Balázs et al. Res Math Sci (2022) 9:48 (m) −m in the rectangle with opposite corners (1, −1), (r, −k). Given ω ∈ H , the map ψ stacks (ω − I{i ≡ m mod k}) copies of the wave i (whenever this is non-negative) −i (m) vertically in increasing order and then removes a point from the bottom of each of the columns 1, 2, ... ,i for each i ≡ m mod k such that ω = 0 (giving the generalised Young −i diagramofanelement of GFP (n)). See Fig. 14 for an example when k = 3(as D ,D ,−m k k earlier, points are labelled as black squares). It is possible to check that these maps are bijections, but as before this is tedious and is left to the reader. The cases for other offset can be proved in the usual fashion, by proving the equalities of generating functions: k(+1) −m f (q) = f (q)q if k = k − m with m ∈{0, 1, ... k − 1}. D ,D ,−m D ,D ,k k k k k These follow naturally from a generalisation of Wright’s bijection and the bijections e o φ , φ of Sect. 3.3, i.e. bijections ,n ,n k( + 1) (m) φ :GFP (n) → GFP n + − m D ,D ,−m D ,D ,k−m ,n k k k k for each  ∈ Z and m ∈{1, 2, ... ,k − 1} as follows. Given an element GFP (n), the D ,D ,−m k k (m) |k−m|(|k−m|+1) map φ adds the points inside the right angled triangle of size attached ,n 2 to either the left or top edge of its generalised Young diagram, depending on whether  ≥ 0 or < 0, respectively. It then uses the new leading diagonal implied by the triangle to k(+1) read off an element of GFP n + − m .See Fig. 15 for examples when D ,D ,k−m k k 2 k = 3. In general not much is known about the functions f (q) for arbitrary k and k . D ,D ,k k k As mentioned earlier, the functions  (q)(= f (q)) are studied in Andrews’ book D ,D ,0 k k [1], and we saw that  (q)and  (q) have product expansions (which gave us product 1 2 expansions for the specialised normalising factors). It turns out that  (q)alsohas a product expansion (proved in Corollary 5.1of[1], assuming Jacobi triple product) and so (3) S (q) can be written as: (3) S (q) =  (q) 2 3 4 5 6 7 8 = 1 + q + 3q + 6q + 11q + 18q + 31q + 49q + 78q + ··· 12i−6 (1 − q ) = . 6i−5 6i−4 2 6i−3 3 6i−2 2 6i−1 12i (1 − q )(1 − q ) (1 − q ) (1 − q ) (1 − q )(1 − q ) i≥1 As in the 2-exclusion case, it would be interesting to know if this could be proved using purely probabilistic methods. It would also be interesting to know whether the other functions f (q)and f (q) are products, as this would imply that the D ,D ,−1 D ,D ,−2 3 3 3 3 (3) (3) normalising factors S (q)and S (q) are products. −1 −2 For k ≥ 4, the functions  (q) are not expected to be products, but can be shown to be sums of products. However, the formulae are very tedious to write down and probably not too illuminating. We finish with another natural question. We have seen that the family of (non- degenerate) nearest neighbour interacting 0-1-2 systems on Z satisfying the blocking measure axioms can be stood up in a uniform way, leading to three variable Jacobi style identities. By specialising to the case of 2-exclusion, we then get two variable identities that are a special case of Theorem 1.3 (setting k = 2). Is there a family of 0-1-···-k processes Balázs et al. Res Math Sci (2022) 9:48 Page 43 of 46 48 ´ ´ MARTON BALAZS, DAN FRETWELL, AND JESSICA JAY (0) 2,23 −→ 42221 644431 0 0 33100 −−−−−− 11 (a) An element of GFP (23) and its image in GFP (32) D ,D ,0 D ,D ,6 3 3 3 3 (1) 1,23 −→ − 542000 76422 2 0 2 211000 −−−−− 00 (b) An element of GFP (23) and its image in GFP (30) D ,D ,−1 D ,D ,5 3 3 3 3 (2) 1,23 + −→ −− 53110 7533 2 1 0 2 2 21100 −−−− 000 (c) An element of GFP (23) and its image in GFP (28) D ,D ,−2 D ,D ,4 3 3 3 3 (0) (1) (2) Fig. 15 Bijections φ , φ and φ 2,23 1,23 1,23 for each k that is nicely behaved (i.e. has blocking measure and can be uniformly stood up) giving a multivariable identity that explains all cases of Theorem 1.3? In general we cannot expect the whole family of 0-1…-k systems on Z to be this well behaved (the stood up processes no longer have nearest neighbour interactions). However, there could be a low-dimensional subfamily of processes for each k that works. Acknowledgements The authors are grateful for discussions with János Engländer regarding the parity of the sum of Bernoulli random variables and for suggestions of an anonymous referee regarding the combinatorial background of our identities. M. Balázs was partially supported by the EPSRC EP/R021449/1 Standard Grant of the UK. On behalf of all authors, the corresponding author states that there is no conflict of interest and that there are no associated data for this manuscript. Appendix A The following are the non-empty sets GFP (n) for 0 ≤ n ≤ 8, whose sizes agree D ,D ,0,m 2 2 with the coefficients in the expansion of S (˜q, t) given in Sect. 3.3. even 48 Page 44 of 46 Balázs et al. Res Math Sci (2022) 9:48 GFP (0) = , D ,D ,0,0 2 2 GFP (1) = , D ,D ,0,2 2 2 GFP (2) = , D ,D ,0,0 2 2 1 0 GFP (2) = , , D ,D ,0,2 2 2 0 1 2 1 0 10 00 GFP (3) = , , , , , D ,D ,0,2 2 2 0 1 2 00 10 11 00 GFP (4) = , , D ,D ,0,0 2 2 00 11 3 2 1 0 20 00 GFP (4) = , , , , , , D2,D2,0,2 0 1 2 3 00 20 GFP (4) = , D ,D ,0,4 2 2 4 3 2 1 0 30 21 11 10 00 00 100 GFP (5) = , , , , , , , , , , , , D ,D ,0,2 2 2 0 1 2 3 4 00 00 10 11 21 30 100 20 10 GFP (5) = , , D ,D ,0,4 2 2 10 20 22 11 00 GFP (6) = , , , D ,D ,0,0 2 2 00 11 22 5 4 3 2 1 0 40 31 20 GFP (6) = , , , , , , , , , D ,D ,0,2 2 2 0 1 2 3 4 5 00 00 11 11 00 00 200 110 100 100 , , , , , , , 20 31 40 100 100 110 200 30 21 20 10 10 GFP (6) = , , , , , D ,D ,0,4 2 2 10 10 20 21 30 6 5 4 3 2 1 0 50 41 32 GFP (7) = , , , , , , , , , , D ,D ,0,2 2 2 0 1 2 3 4 5 6 00 00 00 22 30 21 11 11 10 00 00 , , , , , , , , 10 11 11 21 30 22 32 41 00 300 200 200 110 110 100 , , , , , , , 50 100 200 110 200 110 300 40 31 30 21 20 20 10 10 210 100 GFP (7) = , , , , , , , , , , D ,D ,0,4 2 2 10 10 20 20 21 30 31 40 100 210 33 22 11 00 1100 GFP (8) = , , , , , D ,D ,0,0 2 2 00 11 22 33 1100 7 6 5 4 3 2 1 0 60 51 42 40 GFP (8) = , , , , , , , , , , , , D ,D ,0,2 2 2 0 1 2 3 4 5 6 7 00 00 00 11 31 22 20 11 11 00 00 00 100 220 , , , , , , , , , , 11 20 22 31 40 42 51 60 400 100 211 100 300 200 300 110 100 400 , , , , , , , , 100 211 200 300 110 300 220 100 50 41 32 40 31 30 21 21 30 20 20 GFP (8) = , , , , , , , , , , D ,D ,0,4 2 2 10 10 10 20 20 30 30 21 21 31 40 10 10 10 310 210 210 110 200 100 , , , , , , , , . 32 41 50 100 200 110 210 210 310 The following are the non-empty sets GFP (n) for 0 ≤ n ≤ 8, whose sizes agree D ,D ,−1,m 2 2 with the coefficients in the expansion of S (˜q, t) given in Sect. 3.3. odd Balázs et al. Res Math Sci (2022) 9:48 Page 45 of 46 48 GFP (0) = , D ,D ,−1,1 2 2 − − 0 GFP (1) = , , D2,D2,−1,1 1 00 − − 1 GFP (2) = , , D ,D ,−1,1 2 2 2 00 − 0 GFP (2) = , D ,D ,−1,3 2 2 − − 2 − 0 −00 GFP (3) = , , , , D ,D ,−1,1 2 2 3 00 11 100 − 1 − 0 GFP (3) = , , D ,D ,−1,3 2 2 10 20 − − 3 − 1 −00 −00 GFP (4) = , , , , , D ,D ,−1,1 2 2 4 00 11 200 110 − 2 − 1 − 0 − 0 −10 GFP (4) = , , , , , D ,D ,−1,3 2 2 10 20 21 30 100 − − 4 − 2 − 0 −11 −00 GFP (5) = , , , , , , D ,D ,−1,1 2 2 5 00 11 22 100 300 − 3 − 2 − 1 − 1 − 0 − 0 GFP (5) = , , , , , , D ,D ,−1,3 2 2 10 20 21 30 31 40 −20 −10 −10 −00 , , , , 100 110 200 210 − − 5 − 3 − 1 −11 −11 GFP (6) = , , , , , , D ,D ,−1,1 2 2 6 00 11 22 110 200 −00 −00 −00 −100 , , , , 211 220 400 1100 − 4 − 3 − 2 − 2 − 1 − 1 − 0 − 0 GFP (6) = , , , , , , , , D2,D2,−1,3 10 20 21 30 31 40 32 41 − 0 −30 −21 −20 −20 −10 −00 , , , , , , , 50 100 100 110 200 300 310 −10 GFP (6) = , D ,D ,−1,5 2 2 − − 6 − 4 − 2 − 0 −22 −11 GFP (7) = , , , , , , , D ,D ,−1,1 2 2 7 00 11 22 33 100 300 −00 −00 −00 −200 −110 , , , , , 221 311 500 1100 1100 − 5 − 4 − 3 − 3 − 2 − 2 − 1 − 1 − 1 − 0 GFP (7) = , , , , , , , , , , D ,D ,−1,3 2 2 10 20 21 30 31 40 32 41 50 42 − 0 − 0 −40 −31 −30 −30 −21 −21 , , , , , , , , 51 60 100 100 110 200 110 200 −20 −11 −10 −10 −10 −00 −00 −100 , , , , , , , , 300 210 211 220 400 320 410 2100 −10 −20 GFP (7) = , , , D ,D ,−1,5 2 2 310 210 − − 7 − 5 − 3 − 1 −22 −22 −11 GFP (8) = , , , , , , , D ,D ,−1,1 2 2 8 00 11 22 33 110 200 211 −11 −11 −00 −00 −00 −300 −100 , , , , , , , 220 400 330 411 600 1100 2200 − 6 − 5 − 4 − 4 − 3 − 3 − 2 − 2 − 2 − 1 GFP (8) = , , , , , , , , , D2,D2,−1,3 10 20 21 30 31 40 32 41 50 42 48 Page 46 of 46 Balázs et al. Res Math Sci (2022) 9:48 − 1 − 1 − 0 − 0 − 0 − 0 −50 −41 −32 , , , , , , , , , 51 60 43 52 61 70 100 100 100 −40 −40 −31 −31 −30 −21 −20 −20 , , , , , , , , 110 200 110 200 300 300 211 220 −20 −11 −10 −10 −10 −00 −00 −00 , , , , , , , , 400 310 221 311 500 321 420 510 −210 −200 −110 −100 −100 , , , , . 1100 2100 2100 2 110 3100 −30 −21 −20 −10 −10 GFP (8) = , , , , . D ,D ,−1,5 2 2 210 210 310 320 410 Received: 12 June 2022 Accepted: 15 June 2022 Published online: 21 July 2022 References 1. Andrews, G.: Generalized Frobenius partitions. Am. Math. Soc. 49(301), 66 (1984) 2. Balázs, M.: Growth fluctuations in a class of deposition models. Ann. Inst. H. Poincaré Probab. Stat. 39(4), 639–685 (2003) 3. Balázs, M., Bowen, R.: Product blocking measures and a particle system proof of the Jacobi triple product. Ann. Inst. H. Poincaré Probab. Stat. 54(1), 514–528 (2018) 4. Carinci, G., Giardiná, C., Redig, F., Sasamoto, T.: A generalized asymmetric exclusion process with U (sl ) stochastic q 2 duality. Probab. Theory Relat. Fields 166, 887–933 (2016) 5. Engländer, J., Volkov, S.: Turning a coin over instead of tossing it. J. Theor. Probab. 31(2), 1097–1118 (2018) 6. Liggett, T.: Interacting Particle Systems. Classics in Mathematics, Sprigner, Berlin (2005) 7. Online Encyclopedia of Integer Sequences (A137829). https://oeis.org/A137829 8. Online Encyclopedia of Integer Sequences (A201077). https://oeis.org/A201077 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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