Tightness of Liouville first passage percolation for γ∈(0,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepa ...

Jian Ding; Julien Dubédat; Alexander Dunlap; Hugo Falconet

doi:10.1007/s10240-020-00121-1

Tightness of Liouville first passage percolation for γ∈(0,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepa ...

Ding, Jian; Dubédat, Julien; Dunlap, Alexander; Falconet, Hugo 2020-11-04 00:00:00 TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) by JIAN DING ,JULIEN DUBÉDAT ,ALEXANDER DUNLAP , and HUGO FALCONET ABSTRACT We study Liouville ﬁrst passage percolation metrics associated to a Gaussian free ﬁeld h molliﬁed by the two- dimensional heat kernel p in the bulk, and related star-scale invariant metrics. For γ ∈ (0, 2) and ξ = ,where d is t γ the Liouville quantum gravity dimension deﬁned in Ding and Gwynne (Commun. Math. Phys. 374:1877–1934, 2020), −1 ξ p ∗h we show that renormalized metrics (λ e ds) are tight with respect to the uniform topology. We also show that t∈(0,1) subsequential limits are bi-Hölder with respect to the Euclidean metric, obtain tail estimates for side-to-side distances, and derive error bounds for the normalizing constants λ . CONTENTS 1. Introduction and main statement ........................................... 353 1.1. Strategy of the proof and comparison with previous works ........................... 357 2. Description and comparison of approximations ................................... 359 2.1. Basic properties of φ and ψ ........................................... 360 δ δ 2.2. Comparison between φ and ψ ......................................... 365 δ δ 2.3. Length observables ................................................ 366 2.4. Outline of the proof and roles of φ and ψ ................................... 367 δ δ 3. Russo–Seymour–Welsh estimates ........................................... 367 3.1. Approximate conformal invariance ....................................... 367 3.2. Russo–Seymour–Welsh estimates ......................................... 371 4. Tail estimates with respect to ﬁxed quantiles ..................................... 376 5. Concentration ...................................................... 380 5.1. Concentration of the log of the left–right crossing length ............................ 380 5.2. Weak multiplicativity of the characteristic length and error bounds ...................... 388 5.3. Tightness of the log of the diameter ....................................... 391 5.4. Tightness of the metrics .............................................. 393 Acknowledgements ..................................................... 396 Publisher’s Note ...................................................... 396 Appendix A . ........................................................ 396 A.1. Comparison with the GFF molliﬁed by the heat kernel ............................. 396 A.2. Approximations for δ ∈ (0, 1) .......................................... 400 References ......................................................... 401 1. Introduction and main statement We consider the problem of rigorously constructing a metric for Liouville quantum γ h gravity (LQG), a random geometry formally given by reweighting Euclidean space by e , where h is a Gaussian free ﬁeld. LQG was originally introduced in the physics literature by Polyakov in 1981 [40]. In its mathematical form, the LQG measure is a special case of Gaussian multiplicative chaos, introduced in [29]. In the last two decades, there has been J. Ding was partially supported by NSF grant DMS-1757479 and an Alfred Sloan fellowship. J. Dubédat was Partially supported by NSF grant DMS-1512853. A. Dunlap was partially supported by an NSF Graduate Research Fellowship. © IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2020 https://doi.org/10.1007/s10240-020-00121-1 354 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET an explosion of interest in the probability community towards rigorously constructing the relevant objects. In particular, the LQG measure was constructed rigorously in the regime γ ≤ 2, via a renormalization procedure, in [21]. Other relevant work in this area includes [2, 3, 41–44, 48]. Much remains open regarding the construction of the LQG metric. When γ = 8/3, LQG is intimately connected with the Brownian map [31, 32, 35]and ametric for LQG has been constructed in [36–38]. Substantial work has also been devoted to understanding the distance exponents for natural discrete LQG metrics; see [1, 11, 12, 15, 27, 28]. In [14, 16] some non-universality results were established for ﬁrst-passage γφ percolation distance exponents for metrics of the form e ds,where φ is discretization of a log-correlated Gaussian ﬁeld. This indicates that precisely understanding such expo- nents must involve rather ﬁne information about the structure of the particular ﬁeld in question. The present study concerns the tightness of Liouville ﬁrst-passage percolation (LFPP) metrics, which are natural smoothed LQG metrics. This proves the existence of subsequential limiting metrics. Given this, it remains to show that such limiting metrics are unique in law for each γ ∈ (0, 2) in order to complete the construction of the LQG metric in this regime. After this paper was posted, the latter task was carried out in the series of works [19, 23–26], thus completing the construction. The present study follows three main tightness results for discretized or smoothed LQG metrics. In [9], tightness of LFPP metrics (on a discrete lattice) was proved in the small noise regime for which γ is very small. In [18], tightness was shown for metrics arising in the same way from -scale invariant ﬁelds, still in the small noise regime. In [10], tightness was shown for all γ< 2 for the Liouville graph distance, which is a graph metric equal to the least number of Euclidean balls of a given LQG measure necessary to cover a path between a pair of points. We consider a smoothed Gaussian ﬁeld (1.1) φ (x) := π p (x − y)W(dy, dt) 2 2 δ R |x−y| 2 1 − 2t for x ∈ R and δ ∈ (0, 1),where p (x − y) := e and W is a space–time white 2π t noise. This approximation is natural since it can be uniformly compared on a compact domain with a Gaussian free ﬁeld h molliﬁed by the heat kernel deﬁned on a slightly larger domain, viz. φ and p ∗ h (where ∗ denotes the convolution operator) are comparable. t t/2 Furthermore, this approximation provides some nice invariance and scaling properties on the full plane. For γ ∈ (0, 2),wewilluse thenotation (1.2) ξ := γ/d where d is the “Liouville quantum gravity dimension” deﬁned in [12]. It is known (see Theorem 1.2 and Proposition 1.7 in [12]) that the function γ → γ/d is strictly increasing γ TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 355 and continuous on (0, 2). Therefore, in this article we will be interested in the range − − ξ ∈ (0,(2/d ) ),where (2/d ) = lim γ/d . 2 2 γ ↑2 γ ξφ We consider the length metric e ds (equivalently, the metric whose Riemannian 2ξφ 2 2 metric tensor is given by e ds ), restricted to the unit square [0, 1] .Werecallthat a length metric is a metric such that the distance between two points is given by the inﬁmum over the arc lengths of paths connecting the two points. We denote by λ the 2 ξφ median of the left–right distance of [0, 1] for the metric e ds. Our main theorem is the following. −1 ξφ Theorem 1.—1. If γ ∈ (0, 2), then λ e ds is tight with respect to the δ∈(0,1) 2 2 + uniform topology on the space of continuous functions [0, 1] ×[0, 1] → R .Furthermore, any subsequential limit is almost surely bi-Hölder with respect to the Euclidean metric on [0, 1] . 2. Let K =[0, 1] . If h is a Gaussian free ﬁeld with zero boundary conditions on a bounded open domain D containing K (extended to zero outside of D), then the internal metrics ξ p ∗h −1 √ 2 (λ e ds) on K are tight with respect to the uniform topology of continuous func- δ∈(0,1) tions K × K → R . Furthermore, the normalizing constants (λ ) satisfy δ δ∈(0,1) O | log δ| 1−ξ Q (1.3) λ = δ e where Q = + . γ 2 A year after our article was posted, the subsequent work [13]provedasimilar result to ours when ξ ≥ (2/d ) . However, in that case the tightness does not hold in the uniform topology and the Beer topology on lower semicontinuous functions was used. In order to establish the tightness of the family of renormalized metrics −1 ξφ (d ) := (λ e ds) , we prove a number of uniform estimates for that fam- φ δ∈(0,1) δ∈(0,1) δ δ ily (which also hold when the approximation is the GFF molliﬁed by the heat kernel). Such estimates that are closed under weak convergence also apply to subsequential lim- its. Let us summarize these properties. Let D denote the family of laws of d , δ ∈ (0, 1) 2 2 (i.e. seen as random continuous functions on ([0, 1] ) ), and D denotes its closure under weak convergence (i.e., D also includes the laws of all subsequential limits). 1. Under any P ∈ D, d is P-a.s. a length metric. This is clear for the renormalized metrics d by deﬁnition, and the property of being a length metric extends to limits. (See [6, Exercise 2.4.19].) 2. If d is a metric on R and R is a rectangle, we denote by d(R) the left–right length of R for d . We have the following tail estimates. There exists c, C > 0 356 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET such that for s > 2, uniformly in P ∈ D we have 2 2 −Cs −s −cs (1.4) ce ≤ P d(R) ≤ e ≤ Ce , −c −Cs s log s (1.5) ce ≤ P (d(R) ≥ e ) ≤ Ce . The upper bounds are proved in Section 4, while the lower bounds are conse- quences of the Cameron–Martin theorem, considering shifts of the ﬁeld at the coarsest scale as in [18, Section 5.4]. 3. If d is a metric on R and R is a rectangle, we denote by Diam(R, d) the diam- eter of R for d . We have the following uniform ﬁrst moment bound: (1.6)sup E (Diam(R, d)) < ∞. P∈D This is shown in the course of the proof of Proposition 27 below. 4. Under any P ∈ D, d is P-a.s. bi-Hölder with respect to the Euclidean metric and we have the following bounds for exponents: for α< ξ(Q − 2), β> ξ(Q + 2), and R a rectangle, the families |x − x | d(x, x ) (1.7) sup and sup d(x, x ) |x − x | x,x ∈R x,x ∈R L(d)∈D L(d)∈D are tight. Here L(d) means the law of d . These properties are shown in Propo- sition 28 below. Let us also mention that subsequential limits are consistent with the Weyl scal- ing: for a function f in the Cameron–Martin space of the Gaussian free ﬁeld h,for any coupling (h, d) associated to a subsequential limit of the sequence of laws of ξ p ∗h −1 ξ f √ 2 ((h,λ e ds)) , the couplings (h, d) and (h + f , e · d) are mutually absolutely con- δ>0 tinuous with respect to each other and the associated Radon–Nikodým derivative is the one of the ﬁrst marginal. This can be proved using similar arguments to those of [18, Section 7]. An analogue of this property for the Liouville measure together with the con- servation of the Liouville volume average is enough to characterize the Liouville measure, as seen by Shamov in [48]. It may be interesting to draw a parallel between our work and those in random planar maps, since both aim to obtain the scaling limits of random metrics and the lim- iting objects are related for γ = 8/3. We start with a brief overview on the conver- 1/4 gence of random planar maps. Chassaing and Schaeffer [7] identiﬁed n as the proper scaling and compute certain limiting functionals for random quadrangulations. Marck- ert and Mokkadem [34] established limit theorems (in a sense weaker than Gromov– Hausdorff) and introduced the Brownian map. Le Gall [30] showed tightness for rescaled 2p-angulations in the Gromov–Hausdorff topology and shows that the limiting topology TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 357 is the same as the Brownian map. Le Gall and Paulin [33] showed that the limiting topol- ogy is that of the 2-sphere and Le Gall studied properties of geodesics in [31]. Finally, Le Gall [32] (resp. Miermont [35]) proved the uniqueness of subsequential limits of uniform triangulations and 2p-angulations (resp. quadrangulations). In both cases, the proof relied on a careful study of geodesics and in particular on conﬂuence properties, together with a rough bound quantifying an approximate equivalence of the two metrics to match. In our framework, an important result was obtained in [12], where the authors identiﬁed the exponent of LFPP distances to be 1 − ξ Q + o(1). By contrast, in the ran- 1/4 dom planar map setting, the normalization is exactly n and Chassaing and Schaeffer [7] obtained the convergence in law of some observables. In our case, the tightness of any observable renormalized by its median is far from obvious. This is a common feature of some concentration problems for extrema of random ﬁelds. As an analogy, one can consider the problem of the tightness of the maxima of branching random walks (BRW), where, also by subadditivity, the expected value of the maxima of BRW on a d -ary tree ∗ ∗ at level n is of order (x + o(1))n for some x (which depends on the rate function of the distribution of the increments). A powerful and well-understood method in proving tight- ness for BRW is by an explicit truncated second moment estimate which computes the expected maxima up to additive O(1) constant (see [4] for maxima of BRW and [5]for maxima of discrete GFF). In contrast, in our setup, explicit computation on the distance seems really difﬁcult; in fact it remains a major challenge to compute the value of the dis- tance exponent, let alone computing the distance up to constant. In order to circumvent this difﬁculty, we had to build our proof by exploring delicate intrinsic structure of the distance. We point out here that it was shown in [1]thatfor γ ∈ (0, 2),1 − ξ Q ≥ 0(and it is believed to be > 0), therefore the normalization λ should be thought as small. Furthermore, in our setting where the metrics are on a compact subset of C,we can directly use the uniform topology instead of working with the Gromov–Hausdorff topology (note that the former is stronger than the latter). In this paper, we show tight- ness for the full subcritical range γ ∈ (0, 2) of renormalized side-to-side crossing lengths, point-to-point distance and metrics. Limiting metrics are bi-Hölder with respect to the Euclidean metric. 1.1. Strategy of the proof and comparison with previous works. — In contrast with previous works on the LQG measure, the variational problem deﬁning the LQG metric means that most direct computations are impossible, and in particular most of techniques used in the theory of Gaussian multiplicative chaos and LQG measure are unavailable. This necessitates the more intricate multiscale geometric arguments that we employ. Our tightness proof relies on two key ingredients, a Russo–Seymour–Welsh argu- ment and multiscale analysis. In both parts we extend and reﬁne many arguments used in the previous works [9, 10, 18] on the tightness of various types of LQG metrics. Russo–Seymour–Welsh. — The RSW argument relates, to within a constant factor, quan- tiles of the left–right LFPP crossing distances of a “portrait” rectangle and of a “land- 358 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET scape” rectangle. (By a crossing distance we simply mean the distance between two op- posite sides of a rectangle.) In [9, 10], these crossings are referred to as “easy” and “hard” respectively. The utility of such a result is that crossings of larger rectangles necessarily induce easy crossings of subrectangles, while hard crossings of smaller rectangles can be glued together to create crossings of larger rectangles. Thus, multiscale analysis argu- ments can establish lower bounds in terms of easy crossings and upper bounds in terms of hard crossings. RSW arguments then allow these bounds to be compared. RSW arguments originated in the works [45–47] for Bernoulli percolation, and have since been adapted to many percolation settings. The work [9] introduced an RSW result for LFPP in the small noise regime based on an RSW result for Voronoi percolation devised by Tassion [49]. Tassion’s result is beautiful but intricate, and becomes quite complex when it is adapted to take into account the weights of crossing in the ﬁrst-passage percolation setting, as was done in [9]. The RSW approach of this paper is based on the much simpler approach intro- duced in [18], which relies on an approximate conformal invariance of the ﬁeld. (We recall that the Gaussian free ﬁeld is exactly conformally invariant in dimension 2, and that the LQG measure enjoys an exact conformal covariance.) Roughly speaking, the conformal invariance argument relies on writing down a conformal map between the portrait and landscape rectangles, and analyzing the effect of such a map on crossings of the rectangle. We note that the approximate conformal invariance used in this paper relies in an important way on the exact independence of different “scales” of the ﬁeld, which is manifest in the independence of the white noise at different times in the expres- sion (1.1). Thus, the argument we use here is not immediately applicable to molliﬁcations of the Gaussian free ﬁeld by general molliﬁers (for example, the common “circle-average approximation” of the GFF). The RSW argument of [18] was also adapted in [10]tothe Liouville graph distance case. Tail estimates. — Once the RSW result is established, we derive tail estimates with respect to ﬁxed quantiles. The lower tail estimate is unconditional, while the upper tail estimate depends on a quantity measuring the concentration at the current scale, which will later be uniformly bounded by an inductive argument. Multiscale analysis. — With RSW and tail estimates in hand, we turn to the multiscale analysis part of the paper. This argument turns on the Condition (T) formulated in (5.2) below, which, informally, states that the arclength of the crossing is not concentrated on a small number of subarcs of small Euclidean diameter. The argument of [10] requires similar input, which is a key role of the subcriticality γ< 2. While [10] relies directly on certain scaling symmetries of the Liouville graph distance to use subcriticality, the present work relies on the characterization of the Hausdorff dimension d obtained in [12], along with some weak multiplicativity arguments and concentration obtained from percolation arguments. TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 359 Condition (T). — Our formulation of Condition (T), which has not appeared in previous works, precisely captures the property of the metric needed to obtain the tightness of the left–right crossing distances, the existence of the exponent, and the tail estimates (via a uniform bound on the ). Condition (T) makes sense for LFPP with any underlying ﬁeld and any parame- ter ξ . In particular, this condition or a variant thereof could possibly hold for LFPP for some ξ> 2/d . Therefore, a byproduct of the present work is a simple criterion (that im- plies, as noted above, tightness of the crossing distances, existence of exponents, and tail estimates) that may be applicable more generally. The utility of Condition (T) is that it allows us to use an Efron–Stein argument to obtain a contraction in an inductive bound on the crossing distance logarithm variance. Informally, since the crossing distance feels the effect of many different subboxes, the subbox crossing distances are effectively being averaged to form the overall crossing dis- tance. This yields a contraction in variance. (Of course, the coarse scales also contribute to the variance, and hence the variance of the crossing distance does not decrease as the discretization scale decreases but rather stays bounded.) The way we verify Condition (T) is quite rough: we bound the ﬁeld uniformly over a coarse grained geodesic by the supremum of the ﬁeld over the unit square. It turns out that this bound together with the identiﬁcation of the exponent 1 − ξQis enough to establish the condition. Tightness of the metrics. — Once the tightness of the left–right crossing distance is estab- lished, we turn to the tightness of the diameter and of the metric itself. This is done by a chaining argument, and requires again ξ< 2/d . The diameter is not expected to be tight when ξ> 2/d , since there are points that become inﬁnitely distant from the bulk of the space as the discretization scale goes to 0. 2. Description and comparison of approximations We recall that a white noise W on R is a random Schwartz distribution such that for every smooth and compactly supported test function f , W, f is a centered Gaussian variable with variance f 2 d (see e.g. [8]). The main approximation of the Gaussian L (R ) free ﬁeld that we consider in this paper is deﬁned for δ ∈ (0, 1) by (2.1) φ (x) := π p (x − y)W(dy, dt) 2 2 δ R |x−y| 1 − 2 2t where p (x − y) := e and W is a space–time white noise on [0, 1]× R . This 2π t approximation is different than the one considered in [18] which is x − y −3/2 φ (x) := k t W(dy, dt) 2 t δ R 360 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET for a smooth nonnegative bump function k, radially symmetric and with compact sup- port. Up to a change of variable in t, the difference is essentially replacing p by k.Both ﬁelds are normalized in such a way that E(φ (x)φ (y)) =− log |x − y|+ g(x, y) with g 0 0 continuous (see e.g. Section 2 in [18]): this is the reason for the factor π in (2.1). Let us mention that -scale invariant Gaussian ﬁelds with compactly-supported bump function k 1. are invariant under Euclidean isometries, 2. have ﬁnite-range correlation at each scale, 3. and have convenient scaling properties. The Gaussian ﬁeld φ introduced above satisﬁes 1 and 3 but not 2. Because of the lack of ﬁnite-range correlation, we will also use a ﬁeld ψ (deﬁned in the next section) which satisﬁes 1 and 2 such that sup φ − ψ has Gaussian tails, where we use 0,n 0,n ∞ 2 n≥0 L ([0,1] ) −n the notation φ for φ with δ = 2 . 0,n δ 2.1. Basic properties of φ and ψ . δ δ Scaling property of φ .— We use the scale decomposition −2n φ := φ where φ (x) = π p (x − y)W(dy, dt) n n −2(n+1) 2 2 R n≥0 If we denote by C the covariance kernel of φ ,so C (x, x ) = E(φ (x)φ (x )),thenwe n n n n n have −2n 1 |x−x | − n n 2t C (x, x ) = e dt = C (2 x, 2 x ). n 0 −2(n+1) 2t Therefore, the law of (φ (x)) 2 is the same as (φ (2 x)) 2 . Because of the above, n x∈[0,1] 0 x∈[0,1] 2t 2 −1 we choose δ and not δ in (2.1) so that the pointwise variance φ is log δ . Similarly, for 0 < a < b and x ∈ R , set (2.2) φ (x) := π p (x − y)W(dy, dt) a,b 2 2 a R (d) and note that we have the scaling identity φ (r·) = φ (·). Indeed, we have a,b a/r,b/r t t E(φ (rx)φ (rx )) = π p (rx − y)p (y − rx )dydt a,b a,b 2 2 2 2 a R 2 2 b b 2 2 1 r |x−x | 2t = π p (r(x − x ))dt = e dt, 2 2 2t a a TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 361 and by the change of variable t = r u, this gives 2 2 b (b/r) 2 2 2 1 r |x−x | 1 |x−x | − − 2t 2t e dt = e dt = E(φ (x)φ (x )). a/r,b/r a/r,b/r 2 2t 2 2t a (a/r) −n −k We will use the notation φ when a = 2 and b = 2 for 0 ≤ k ≤ n. k,n Maximum and oscillation of φ .— We have the same estimates for the supremum of the ﬁeld φ as those for the -scale invariant case considered in [18] (it is essentially a union 0,n bound combined with a scaling argument). The following proposition corresponds to Lemma 10.1 and Lemma 10.2 in [18]. Proposition 2 (Maximum bounds).— We have the following tail estimates for the supremum of φ over the unit square: for a > 0,n ≥ 0, 0,n √ a − n log 4 (2.3) P max |φ |≥ a(n + C n) ≤ C4 e 0,n [0,1] as well as the following moment bound: if γ< 2, then γ max |φ | γ n+O( n) 2 0,n [0,1] (2.4) E(e ) ≤ 4 We will also need some control on the oscillation of the ﬁeld φ . We introduce 0,n ∞ d d the following notation for the L -norm on a subset of R . If A is a subset of R and f : A → R , we set (2.5) f := sup |f (x)| x∈A We introduce the following notation to describe the oscillation of a smooth ﬁeld φ:if A ⊂ R we set (2.6)osc (φ) := diam(A) ∇φ , so that if A is convex then sup |φ(x) − φ(y)|≤ osc (φ) and x,y∈A −n max osc (φ ) ≤ C2 ∇φ , P 0,n 0,n [0,1] P∈P ,P⊂[0,1] where P denotes the set of dyadic blocks at scale n,viz. −n (2.7) P := {2 ([i, i + 1]×[j, j + 1]) : i, j ∈ Z}. In order to simplify the notation P ∈ P ,P ⊂[0, 1] later on, we also set 1 2 (2.8) P := {P ∈ P : P ⊂[0, 1] }. n 362 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Proposition 3 (Oscillation bounds).— We have the following tail estimates for the oscillation of φ : there exists C > 0, σ > 0, so that, for all x,ε > 0,n ≥ 0, 0,n −n n 2σ (2.9) P 2 ∇φ ≥ x ≤ C4 e 0,n 2 [0,1] as well as the following moment bound: for a > 0, there exists c > 0 so that for n ≥ 0, ε −n +ε 2ε an 2 ∇φ 2 0,n c n +O(n ) 2 a [0,1] (2.10) E e ≤ e Proof. — Inequality (2.9) was obtained between Equation (10.3) and Equation ε −n (10.4) in [18]. Now, we prove (2.10). Set a := an ,O = 2 ∇φ ,and take n n 0,n [0,1] 2 α x = a σ + ασ n with α> 0so that = log 4. We have, using (2.9), n n ∞ ∞ ∞ a O a O s s n 2 s n n n n 2a σ P e ≥ x dx = P e ≥ e e ds ≤ C4 e e ds a x n n e a x a x n n n n By a change of variable (s ↔ a σ s + (a σ) ), we get n n ∞ 2 ∞ ∞ s 2 2 − 1 2 2 s 1 2 2 s 2 s a σ − a σ − 2a σ 2 n 2 2 n 2 e e ds = a σ e e ds = a σ e e ds n n xn a x −a σ α n n n n 2 2 2 −bx −1 −ba since x = a σ + ασ n. Using that e dx ≤ (2ab) e ,weget n n 2ε a O O(n ) a O a x n n n n n n P e ≥ x dx ≤ e . The result follows from writing E(e ) ≤ e + a x n n a O n n P(e ≥ x)dx. a x n n Deﬁnition of ψ .— We ﬁx a smooth, nonnegative, radially symmetric bump function such that 0 ≤ ≤ 1and is equal to one on B(0, 1) and to zero outside B(0, 2).We also ﬁx small constants r > 0and ε > 0. We will specify these constants later on. In 0 0 particular, ε appears in the main proof in (5.11) and its ﬁnal effect is in (5.16). All other constants C, c will implicitly depend on r and ε . Then, we introduce for each δ ∈[0, 1], 0 0 the ﬁeld 1 1 Tr (2.11) ψ (x) := (x−y)p (x−y)W(dy, dt) = p (x − y)W(dy, dt) δ σ 2 2 2 2 δ R δ R ε Tr where σ = r t| log t| , (·) := (·/σ ) and p := p t . t 0 σ t σ t t Thanks to the truncation, the ﬁelds (ψ ) have ﬁnite correlation length δ δ∈[0,1] 8r sup t| log t| . t∈[0,1] TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 363 Decomposition in scales and blocks of ψ .— We have the scale decomposition Tr (2.12) ψ(x) := p (x − y)W(dy, dt) 0 R −2k+2 Tr = p t (x − y)W(dy, dt) = ψ (x) k,P −2k 2 P k=1 k≥1 P∈P P∈P k k −2k+2 Tr where ψ is deﬁned for P ∈ P by ψ (x) := p (x − y)W(dy, dt) and thus has k,P k k,P −2k 2 P ε −k correlation length less than Ck 2 . In particular, a ﬁxed block ﬁeld is only correlated 2ε with fewer than Ck other block ﬁelds at the same scale. In fact, when we apply the Efron–Stein inequality (see (5.9)) we will use the following decomposition: (2.13) ψ = ψ + ψ (x) 0,n 0,K K,n,P P∈P −2K+2 Tr where ψ (x) := p (x − y)W(dy, dt). K,n,P −2n 2 P We note that there is a formal conﬂict in notation between (2.2)and (2.13), butitwill always be clear from context whether the second subscript is a number or an element of P (a set), so confusion should not arise. Variance bounds for φ and ψ .— Later on we will need the following lemma. δ δ Lemma 4. — There exists C > 0 so that for δ ∈[0, 1] and x, x ∈ R , we have |x − x | (2.14)Var φ (x) − φ (x ) + Var ψ (x) − ψ (x ) ≤ C . δ δ δ δ −z Proof. — We start by estimating the ﬁrst term. Using the inequality 1 − e ≤ z ≤ √ √ −z z for z ∈[0, 1] and 1 − e ≤ 1 ≤ z for z ≥ 1we get t t t t Var φ (x) − φ (x ) = C p ∗ p (0) − p ∗ p (x − x ) dt δ δ 2 2 2 2 = C p (0) − p (x − x ) dt t t 1 |x−x | 2t = C (1 − e )dt 2 t dt |x − x | ≤ C|x − x | = C . 3/2 2 t δ δ 364 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Similarly, for the second term, we have Tr Tr Tr Tr Var ψ (x) − ψ (x ) = C p t ∗ p t (0) − p t ∗ p t (x − x ) dt. δ δ 2 2 2 2 Tr Tr Set p ∗ p =: p (x)q (x). Using the identity p (y)p (x − y) = p (x)p (y − x/2) we get t t t t t/2 t/2 t t/4 2 2 p (y)p (x − y) t/2 t/2 q (x) = (y) (x − y)dy t σ σ t t 2 p (x) = p (y − x/2) (y) (x − y)dy. t/4 σ σ t t We rewrite the variance in terms of q : replacing x − x by z we look at Var ψ (x) − ψ (x ) δ δ = C (p (0)q (0) − p (z)q (z))dt t t t t 1 1 = C p (0)(q (0) − q (z))dt + C q (z)(p (0) − p (z))dt. t t t t t t 2 2 δ δ We deal with these two terms separately. For the second one, since 0 ≤ ≤ 1, we have 0 ≤ q ≤ 1. Therefore, following what we did for φ above we directly have t δ |z| −1 0 ≤ q (z)(p (0) − p (z))dt ≤ C . For the ﬁrst term, since p (0) = Ct ,itisenough 2 t t t t to get the bound t|q (0) − q (z)|≤ C|z| to complete the proof of the lemma. Changing t t variables, we have √ √ −2|y| q (z) = C e ( ty + z/2) ( ty − z/2)dy. t σ σ t t Therefore, using that 0 ≤ ≤ 1, −2|y| |q (z) − q (0)|≤ C e | ( ty + z/2) − ( ty)|dy t t σ σ t t −2|y| + C e | ( ty − z/2) − ( ty)|dy σ σ t t 2 |z| 2 −2|y| −2|y| ≤ C|z| e ∇ dy ≤ C ∇ 2 e dy. σ 2 R 2 σ 2 R t R ε t Since σ = r t| log t| , we see that sup < ∞, and the result follows. t 0 t∈[0,1] t TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 365 2.2. Comparison between φ and ψ .— The following proposition justiﬁes the intro- δ δ duction of the ﬁeld ψ . Proposition 5. — There exist C > 0 and c > 0 such that for all x > 0, we have −cx (2.15) P sup φ − ψ ≥ x ≤ Ce . 0,n 0,n [0,1] n≥0 Proof. — For k ≥ 1, we introduce the quantity D (x) := φ (x) − ψ (x).The k k−1,k k−1,k proof follows from an adaptation of Lemma 2.7 in [17] as soon as we have the estimates 2ε −ck (2.16)VarD (x) ≤ Ce and (2.17)Var φ (x) − φ (y) + Var ψ (x) − ψ (y) ≤ 2 |x − y|. k k k k (The estimate (2.16) is weaker than that used in [17, Lemma 2.7] but still much stronger than required for the proof given there.) Note that (2.17) follows from Lemma 4 and for (2.16) we proceed as follows: ﬁrst note that −2k+2 2 2 E φ (x) − ψ (x) = p (y) (1 − (y)) dydt. k−1,k k−1,k σ −2k 2 2 R −1 −σ /t For every y,wehave p (y)(1 − (y)) ≤ (2π t) e since 0 ≤ ≤ 1and (y) = 1 t/2 σ σ σ t t t for |y|≤ σ . Therefore, −2k+2 2 2ε −ck E φ (x) − ψ (x) ≤ p (y)dydt ≤ Ce . k−1,k k−1,k −2k 2π t 2 2 R Let us point out that in fact E( φ − ψ )< ∞ holds but we n,n+1 n,n+1 n≥0 [0,1] won’t use it. Since we will be working with two different approximations of the Gaus- sian free ﬁeld, we introduce here some notation, referring to one ﬁeld or the other. We will denote by R := [0, a]×[0, b] the rectangle of size (a, b).Wedeﬁne a,b (2.18)X := sup φ − ψ a,b 0,n 0,n a,b n≥0 and X := X for the supremum norm of the difference between the two ﬁelds on vari- a a,a ous rectangles. 366 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET (n) (n) 2.3. Length observables. — The symbol L (φ) (and similarly L (ψ )) will refer to a,b a,b ξφ 0,n the left–right distance of the rectangle R for the length functional e ds: a,b (n) ξφ 0,n (2.19)L (φ) := inf e ds, a,b where ds refers to the Euclidean length measure and the inﬁmum is taken over all smooth curves π connecting the left and right sides of R . We will sometimes consider a geodesic a,b associated to this variational problem. Such a path exists by the Hopf–Rinow theorem and a compactness argument. We introduce some notation for the quantiles associated to this observable: (n) (n) (n) (n) (φ, p) (similarly (ψ, p))issuchthat P L (φ) ≤ (φ) = p. For high quantiles, a,b a,b a,b a,b (n) (n) (n) we introduce (φ, p) := (φ, 1 − p).Notethat (φ, p) is increasing in p whereas a,b a,b a,b (n) (φ, p) is decreasing in p. Note that both are well-deﬁned, i.e., there are no Dirac deltas a,b (n) in the law of L . This follows from an application of the Cameron–Martin formula. We a,b will also need the notation (φ, p) (2.20) (φ, p) := max k≤n (φ, p) (k) (k) ¯ ¯ where (φ, p) := (φ, p) and (φ, p) := (φ, p). k k 1,1 1,1 The following inequalities are straightforward: −ξ X (n) (n) ξ X (n) a,b a,b (2.21) e L (ψ ) ≤ L (φ) ≤ e L (ψ ) a,b a,b a,b Therefore, using Proposition 5 (and a union bound, if necessary), we obtain that for some C > 0 (depending only on a and b), for any ε> 0we have √ √ (n) (n) (n) −ξ C | log ε/C| ξ C | log ε/C| ¯ ¯ ¯ e (ψ, p + ε) ≤ (φ, p) ≤ e (ψ, p − ε) a,b a,b a,b √ √ (n) (n) (n) −ξ C | log ε/C| ξ C | log ε/C| e (ψ, p − ε) ≤ (φ, p) ≤ e (ψ, p + ε) a,b a,b a,b In particular, there exists C > 0 such that, uniformly in n, −1 ¯ ¯ (ψ, p/2) ≥ C (φ, p), (ψ, p/2) ≤ C (φ, p) and n p n n p n (2.22) (ψ, p/2) ≤ C (φ, p). n p n Now, we discuss how the scaling property of the ﬁeld φ translates at the level of lengths. We will use the following equality in law: for a, b > 0and 0 ≤ m ≤ n, (d) (m,n) (n−m) −m (2.23)L (φ) = 2 L (φ). m m a,b a·2 ,b·2 TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 367 (n) Finally, for a rectangle P with two marked opposite sides, we deﬁne L (P,φ) to ξφ 0,n be the crossing distance between the two marked sides under the ﬁeld e . The marked sides will be clear from context: if we call P a “long rectangle,” then we mean that the (n) marked sides are the two shorter sides, so that L (P,φ) is the distance across P “the long way.” 2.4. Outline of the proof and roles of φ and ψ .— The key idea of the proof is to δ δ obtain a self-bounding estimate associated to a measure of concentration of some ob- servables, say rectangle crossing lengths. This is naturally expected because of the tree structure of our model. We introduce a general condition, which we call Condition (T), (see (5.2)) which ensures a contraction in the self-bounding estimate (5.19), which relates a measure of concentration at scale n, the variance, with the measure of concentration that we inductively bound, (see (2.20)), which is at a smaller scale. n−K We then prove that this condition, which depends only on ξ and on the ﬁeld con- sidered, is satisﬁed when ξ ∈ (0,(2/d ) ). This proof uses a result taken from [12] about the existence of an exponent for circle average Liouville ﬁrst passage percolation and this is the reason we don’t consider the simpler -scale invariant ﬁeld with compactly- supported kernel but the ﬁeld φ , which can be compared to the circle average process by a result obtained in [11]. The roles of φ and ψ in the proof are the following. δ δ 1. Prove Russo–Seymour–Welsh estimates for φ. 2. Prove tail estimates w.r.t. low and high quantiles for both φ and ψ : a) Lower tails: Use directly the RSW estimates together with a Fernique-type argument for the ﬁeld ψ with local independence properties. b) Upper tails: use a percolation/scaling argument, percolation using ψ and scaling using φ. 3. Concentration of the log of the left–right distance: use Efron–Stein for the ﬁeld ψ (because of the local independence properties at each scale). This gives the same result for φ. 4. To conclude for the concentration of diameter and metric, this is essentially a chaining/scaling argument using only the ﬁeld φ. 3. Russo–Seymour–Welsh estimates 3.1. Approximate conformal invariance. — In order to establish our RSW result, we ﬁrst show an approximate conformal invariance property of the ﬁeld. The arguments in this section are similar to those of [18, Section 3.1]. The difference is that the Gaussian kernel has inﬁnite support. |x−y| t 2t Recall that φ (x) = p (x − y)W(dy, dt) where p (x − y) = e .Consider δ 2 2 t δ R 2π t a conformal map F between two bounded, convex, simply-connected open sets U and 368 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Vsuch that |F | ≥ 1onU, F < ∞ and F < ∞. (We point out here that the U U assumption |F |≥ 1 will be obtained later on by starting from a very small domain; this is exactly the content of Lemma 11.) We consider another ﬁeld φ (x) = p (x − δ 2 2 δ R ˜ ˜ y)W(dy, dt) where W is a white noise that we will couple with W in order to compare φ and φ ◦ F. The coupling goes as follows: for y ∈ U, t ∈ (0, ∞),let y = F(y) ∈ Vand δ δ 2 2 2 t = t|F (y)| and set W(dy , dt ) =|F (y)| W(dy, dt).Thatis, forevery L function ω on V × (0, ∞), ω(y , t )W(dy , dt ) = ω(F(y), t|F (y)| ) F (y) W(dy, dt) and both sides have variance ω . The rest of the white noises are chosen to be inde- c c pendent, i.e., W ,W and W are jointly independent. |U ×(0,∞) |U×(0,∞) |V ×(0,∞) Lemma 6. — Under this coupling, we can compare the two ﬁelds φ (F(x)) and φ (x) on a δ δ compact, convex subset K of U as follows, (δ) (δ) (3.1) φ (F(x)) − φ (x) = φ (x) + φ (x), δ δ L H (δ) where φ (L for low frequency noise) is a smooth Gaussian ﬁeld whose L -norm on K has uniform (δ) Gaussian tails, and φ (H for high frequency noise) is a smooth Gaussian ﬁeld with uniformly bounded (δ) (δ) pointwise variance (in δ and x ∈ K). Furthermore, φ is independent of (φ ,φ ). H L This aforementioned independence property will be crucial for our argument. Proof. — Step 1: Decomposition. For ﬁxed F and small δ, we decompose φ (x) − (δ) (δ) (δ) φ (F(x)) = φ (x) + φ (x) + φ (x),where 1 2 3 −2 F (y) | | (δ) φ (x) = p (x−y) − p 2 F(x)−F(y) F (y) W(dy, dt) F (y) 2 | | U δ −2 F (y) | | F(x) − F(y) t t = p (x − y) − p W(dy, dt) 2 2 2 F (y) U δ 1 1 (δ) t t φ (x) = p (x − y)W(dy, dt) − p F(x) − y W(dy, dt) 2 2 c 2 c 2 U δ V δ + p t (x − y)W(dy, dt) −2 U F (y) | | F(x) − F(y) (δ) φ (x) =− p W(dy, dt) −2 2 F (y) U δ F (y) | | (δ) (δ) (δ) Remark also that φ is independent of φ , φ ,and φ . 3 1 2 TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 369 (δ) Step 2: Conclusion, assuming uniform estimates. We will estimate φ , i = 1, 2, 3, over K. In what follows, we take x, x ∈ K. We assume ﬁrst the following uniform esti- mates: (δ) (δ) E((φ (x) − φ (x )) ) ≤ C x − x , 1 1 (δ) (δ) (δ) E((φ (x) − φ (x )) ) ≤ C x − x , E φ (x) ≤ C. 2 2 3 An application of Kolmogorov’s continuity criterion and Fernique’s theorem give uni- (δ) (δ) (δ) (δ) (δ) (δ) (δ) form Gaussian tails for φ and φ . We then set φ := φ and φ := φ + φ . 1 2 H 3 L 1 2 Step 3: Uniform estimates. (δ) (δ) First term. We prove that E((φ (x) − φ (x )) ) ≤ C |x − x | by controlling 1 1 F(x) − F(u) t t p (x − y) − p 2 2 F (y) 0 U F(x ) − F(u) t t − p x − y + p dydt 2 2 F (y) |x| − 2 By introducing p(x) = e and by a change of variable t ↔ 2t , it is equivalent (up to a multiplicative constant) to bound from above the quantity dt x − y F(x) − F(y) (3.2) p − p t t tF (y) 0 U x − y F(x ) − F(y) − p + p dy. t tF (y) We will estimate this term by considering the case where t ≤ |x − x | and the case where t ≥ |x − x |. 2 2 1 2 Step 3.(A): Case t ≥ |x − x |. Using the identity |x − y| +|x − y| = |x − x | + x+x 2 −z 2|y − | and the inequality 1 − e ≤ z,weget |x−x | x − y x − y 4t (3.3) p − p dy ≤ Ct (1 − e ) ≤ C x − x . t t Similarly, F(x) − F(y) F(x ) − F(y) (3.4) p − p dy tF (y) tF (y) 2 2 ≤ C F(x) − F(x ) ≤ C x − x , where the constant C depends on F . Then the corresponding part in (3.2)isbounded dt from above by |x − x | ≤ C|x − x |. |x−x | 370 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Step 3.(B): For t ≤ |x − x |, using the Taylor inequality |F(x) − F(y) − F (y)(x − y)|≤ F |x − y| and the mean value inequality (as we have assumed that K is con- vex), x − y F(x) − F(y) (3.5) p − p t tF (y) 2 2 F(x)−F(y) |x − y| |x − y| |x − y| − inf α(x−y)+(1−α) α∈(0,1) F (y) 2t ≤ C + e . t t t Step 3.(B): case (a). If y ∈ B(x,ε) for ε small enough (depending only on F ), we 1 2 have, using again |F(x) − F(y) − F (y)(x − y)|≤ F |x − y| , uniformly in α ∈ (0, 1), F(x) − F(y) 1 1 α(x − y) + (1 − α) ≥|x − y|− F |x − y| ≥ |x − y|. F (y) 2 2 Therefore, for such y’s we have, coming back to (3.5), |x−y| x − y F(x) − F(y) |x − y| 4t p − p ≤ C e . t tF (y) t For this case we get the bound x − y F(x) − F(y) p − p dy t tF (y) B(x,ε) 6 2 |x−y| |x − y| −2 4 2t ≤ C e = Ct E(|B 2 | ) ≤ Ct , B(x,ε) where B denotes a two-dimensional Gaussian variable with covariance matrix t times |x−x | dt 4 the identity. This term contributes to (3.2)as C t ≤ C|x − x |. Step 3.(B): case (b). Now, for t ≤ |x − x | and y ∈ U \ B(x,ε) we write √ √ |x−x | |x−x | dt x − y dt p dy ≤ C P(|B 2 | >ε) t t t 0 U\B(x,ε) 0 |x−x | dt 2t ≤ C e ≤ C x − x , and similarly |x−x | dt F(x) − F(y) p dy ≤ C x − x , t tF (y) 0 U\B(x,ε) −1 where the constant C depends on F and (F ) . U U TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 371 Applying Step 3.(A) and then Step 3.(B) twice (once for x and then again for x )to (δ) (δ) (3.2), we get E((φ (x) − φ (x )) ) ≤ C |x − x |. 1 1 (δ) (δ) | | Second term. We want to prove here that E((φ (x) − φ (x )) ) ≤ C x − x . 2 2 Note that three terms contribute to δφ . The third one is a nice Gaussian ﬁeld indepen- dent of δ. The ﬁrst two terms are similar, so we will just focus on the ﬁrst one, namely (δ) 1 φ (x) := p (x − y)W(dy, dt).Wehave, similarlyto(3.2)and (3.3), c 2 2,1 U δ (δ) (δ) E φ (x) − φ (x ) 2,1 2,1 t t = p (x − y) − p (x − y) dydt 2 2 2 c δ U dt x − y x − y ≤ C p − p dy t c t t 0 U |x−x | dt x − y x − y ≤ C p + p dy + C|x − x |. t c t t 0 U The remaining term can be controlled as follows (noting the symmetry between x and x ): √ √ |x−x | |x−x | |x−y| dt 1 dt 2t e dy ≤ C P(|B 2 | > d) t c t t 0 U 0 |x−x | dt 2t ≤ C e ≤ C x − x , (δ) (δ) c 2 where d = d(K, U ).Thus E((φ (x) − φ (x )) ) ≤ C |x − x |. 2 2 (δ) Third term. We give here a bound on the pointwise variance of φ . By using |x−y| 2 − x−y δ F(x)−F(y) | | (δ) Ct dt e ≥ we get E(φ (x) ) ≤ dy ≤ C. 2 2 F (y) C cδ t R t 3.2. Russo–Seymour–Welsh estimates. — The main result of this section is the follow- ing RSW estimate. It shows that appropriately-chosen quantiles of crossing distances of “long” and “short” rectangles at the same scale can be related by a multiplicative factor that is uniform in the scale. This is the equivalent of Theorem 3.1 from [18] but with the ﬁeld molliﬁed by the heat kernel instead of a compactly-supported kernel. It holds for any ﬁxed ξ> 0. Proposition 7 (RSW estimates for φ ).— If [A, B]⊂ (0, ∞), there exists C > 0 such that a a for (a, b), (a , b ) ∈[A, B] with < 1 < ,for n ≥ 0 and ε< 1/2, we have, b b (n) (n) C log |ε/C| (3.6) (φ, ε/C) ≤ C (φ, ε)e ; a ,b a,b (n) (n) C C log |ε/C| ¯ ¯ (3.7) (φ, 3ε ) ≤ C (φ, ε)e . a ,b a,b 372 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET The following corollary then follows from Propositions 5 and 7. Corollary 8 (RSW estimates for ψ ).— Under the same assumptions as used in Proposition 7, we have (n) (n) C log |ε/C| (3.8) (ψ, ε/C) ≤ C (ψ, ε)e a ,b a,b and (n) (n) C log |ε/C| ¯ ¯ (3.9) (ψ, 3ε ) ≤ C (ψ, ε)e . a,b a ,b We point out that the constants C in (3.8)and (3.9) are not equal to those in (3.6) and (3.7). The remaining parts of the section will only deal with approximations associ- ated with φ so we will omit this dependence in the various observables. We describe below the main lines of the argument. Consider R and R ,two a,b a ,b a a rectangles with respective side lengths (a, b) and (a , b ) satisfying < 1 < . Suppose b b that we could take a conformal map F : R → R mapping the long left and right a,b a ,b sides of R to the short left and right sides of R . (This is not in fact possible since a,b a ,b there are only three degrees of freedom in the choice of a conformal map, but for the sake of illustration we will consider this idealized setting ﬁrst.) Then the proof goes as follows. Take a geodesic π ˜ for φ for the left–right crossing of R . Then, using the cou- 0,n a,b pling (3.1), we have φ φ ξφ (F(π( ˜ t))) 0,n 0,n 0,n L (R ) ≤ L (F(π) ˜ ) = e |F (π( ˜ t))|·|π˜ (t)|dt a ,b ξ(φ +δφ +δφ ) 0,n L H ≤ F e ds a,b π ˜ ξ δφ ˜ L ξ φ ξδφ R 0,n H a,b ≤ F e e e ds. a,b π ˜ ˜ ˜ It is essential that π ˜ is φ measurable and φ is independent of δφ . Then, we can use 0,n 0,n H the following lemma. Lemma 9. — If is a continuous ﬁeld and is an independent continuous centered Gaussian ﬁeld with pointwise variance bounded above by σ > 0, then we have, as long as ε is sufﬁciently small compared to σ , 2 −1 2σ log ε 1. ( + , ε) ≤ e (, 2ε); 1,1 1,1 2 −1 2σ log ε ¯ ¯ 2. ( + , 2ε) ≤ e (, ε). 1,1 1,1 2 −1 Proof. — Fix s := 2σ log ε throughout the proof. Let π() be a geodesic as- sociated with the left–right crossing length for the ﬁeld , and deﬁne the measure μ on TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 373 −1 π() by μ(ds) = L () e ds,so e ds = 1. Conditionally on , using Jensen’s in- 1,1 π() −1 2 equality with α = = (log ε )/(2σ ), which is greater than 1 for small enough ε, 2σ and Chebyshev’s inequality, we have + s α αs (3.10) P e ds > e L () | ≤ P e dμ ≥ e | 1,1 π() π() 1 2 2 α σ −αs 2 2σ ≤ e e = e = ε. To bound from above L ( + ), we take a geodesic for and use the moment estimate 1,1 2 −1 (3.10). We start with the left tail. Still with s := 2σ log ε ,wehave −s P L () ≤ ( + , ε)e 1,1 1,1 s −s ≤ P L ( + ) ≤ e L (), L () ≤ ( + , ε)e 1,1 1,1 1,1 1,1 + P L ( + ) > e L () 1,1 1,1 + s ≤ P L ( + ) ≤ ( + , ε) + P e ds > e L () 1,1 1,1 1,1 π() ≤ 2ε. For the right tail, we have similarly that P L ( + ) ≥ (, ε)e 1,1 1,1 ¯ ¯ ≤ P L ( + ) ≥ (, ε)e , (, ε) ≥ L () 1,1 1,1 1,1 1,1 + P L () ≥ (, ε) 1,1 1,1 ≤ P L ( + ) ≥ e L () + ε ≤ 2ε, 1,1 1,1 which concludes the proof of the lemma. The previous reasoning does not apply directly to rectangle crossing lengths but provides the following proposition. Recall that K is a compact subset of U. Let A,Bbe two boundary arcs of K and denote by L the distance from A to B in K for the metric ξφ 0,n e ds;wedenote A := F(A),B := F(B),K := F(K),and L is the distance from A to ξ φ 0,n B in K for e ds.Recallthatwehave |F |≥ 1 on U. In the application we will achieve this by scaling U to be sufﬁciently small. Proposition 10. — We have the following comparisons between quantiles. There exists C > 0 such that 1. if for some l > 0 and ε< 1/2, P (L ≤ l) ≥ ε,then P (L ≤ l ) ≥ ε/4 with l = C log ε/2C | | F e . 2. if for some l > 0 and ε< 1/2, P (L ≤ l) ≥ 1 − ε, then P (L ≤ l ) ≥ 1 − 3ε with C log ε/2C | | l = F e . K 374 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET FIG.1.—Illustration of Lemma 12 Now, we want to prove a similar result for rectangle crossing lengths. We will need the three following lemmas that were used in [18]. The ﬁrst one is a geometrical con- struction, the second one is a complex analysis result and the last one comes essentially from [39] together with an approximation argument. In these lemmas, by “crossings” we mean continuous path from marked sides to marked sides. Lemma 11 (Lemma 4.8 of [18]).— If a and b are two positive real numbers with a < b, there exists j = j(b/a) and j rectangles isometric to [0, a/2]×[0, b/2] such that if π is a left–right crossing of the rectangle [0, a]×[0, b], at least one of the j rectangles is crossed in the thin direction by a subpath of that crossing. Lemma 12 (Step 1 in the proof of Theorem 3.1 in [18]).— If a/b < 1 and a /b > 1,there exists m, p ≥ 1 and two ellipses E , E with marked arcs (AB), (CD) for E and (A B ), (C D ) for p p E such that: p p 1. Any left–right crossing of [0, a/2 ]×[0, b/2 ] is a crossing of E . 2. Any crossing of E is a left–right crossing of [0, a ]×[0, b ]. 3. When dividing the marked sides of E into m subarcs of equal length, for any pair of such subarcs (one on each side), there exists a conformal map F : E → E and the pair of subarcs is mapped to subarcs of the marked sides of E . 4. For each pair, the associated map F extends to a conformal equivalence U → V where E ⊂ U, E ⊂ V and |F |≥ 1 on U. We refer the reader to Figure 1 for an illustration. Lemma 13 (Positive association and square-root-trick).— If k ≥ 2 and (R ,..., R ) denote 1 k a collection of k rectangles, then, for (x ,..., x ) ∈ (0, ∞) , we have 1 k (n) (n) P L (R )> x ,..., L (R )> x 1 1 k k (n) (n) ≥ P L (R )> x ··· P L (R )> x . 1 1 k k TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 375 An easy consequence of this positive association is the so-called “square-root-trick”: 1/k (n) (n) max P L (R ) ≤ x ≥ 1 − 1 − P ∃i ≤ k : L (R ) ≤ x . i i i i i≤k The main result of this section, Proposition 7, is a rephrasing of the following one. Proposition 14. — We have the following comparisons between quantiles. If a/b < 1 and a /b > 1, there exists C > 0 such that, for any ε ∈ (0, 1/2), (n) (n) C log ε/C | | 1. if P L ≤ l ≥ ε, then P L ≤ Cle ≥ ε/C, a,b a ,b (n) (n) C log ε/C | | 1/C 2. and if P L ≤ l ≥ 1 − ε, then P L ≤ Cle ≥ 1 − 3ε . a,b a ,b Proof. — We provide ﬁrst a comparison between low quantiles and then a compar- ison between high quantiles. (n) Step 1: Comparison of small quantiles. Suppose P(L ≤ l) ≥ ε. By Lemma 11 and a,b (n) (n) union bound, P(L ≤ l) ≥ ε/j . Furthermore, by iterating, we have P(L ≤ l) ≥ p p a/2,b/2 a/2 ,b/2 ε/j . Under this event, by Lemma 12, there exists a crossing of E between two subarcs p 2 of E (one on each side) hence with probability at least ε/(j m ), one of these crossings has length at most l . By the left tail estimate Proposition 10 and Lemma 12,weobtaina C > 0 (depending also on F )suchthatfor all ε, l > 0: p 2 (n) (n) C log ε/(2Cj m ) p 2 | | P L ≤ l ≥ ε ⇒ P L ≤ Cle ≥ ε/(4j m ), a,b a ,b hence the ﬁrst assertion. (n) Step 2: Comparison of high quantiles. Now suppose P(L ≤ l) ≥ 1 − ε.By a,b Lemma 11 (to start with a crossing at a lower scale) and Lemma 13 (square-root-trick), (n) (n) 1/j we have P(L ≤ l) ≥ 1 − ε . Furthermore, by iterating, we have P(L ≤ l) ≥ p p a/2,b/2 a/2 ,b/2 (n) 1/j 1 − ε .Onthe event {L ≤ l}, the ellipse E from Lemma 12 has a crossing of p p a/2 ,b/2 length ≤ l between two marked arcs. Again by subdividing each its marked arcs into m subarcs and applying the square-root trick, we see that for at least one pair of subarcs, −p −2 j m there is a crossing of length ≤ l with probability ≥ 1 − ε . Combining with the right- tail estimate Proposition 10 and Lemma 12,weget: (n) (n) C log ε/C 1/C | | (3.11) P L ≤ l ≥ 1 − ε ⇒ P L ≤ Cle ≥ 1 − 3ε , a,b a ,b which completes the proof. Remark 15. — The importance of the Russo–Seymour–Welsh estimates comes from the following: percolation arguments/estimates work well when taking small quan- tiles associated with short crossings and high quantiles associated with long crossings. Thanks to the RSW estimates, we can instead keep track only of low and high quantiles associated to the unit square crossing, (p) and (p). n n 376 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET 4. Tail estimates with respect to ﬁxed quantiles Lower tails. — This is where we take r small enough (recall the deﬁnition (2.11)) to obtain some small range of dependence of the ﬁeld ψ so that a Fernique-type argument works. Proposition 16 (Lower tail estimates for ψ ).— We have the following lower tail estimate: for p small enough, but ﬁxed, there is a constant C so that for all s > 0, (n) −s −cs (4.1) P L (ψ ) ≤ e (ψ, p) ≤ Ce . 1,3 Proof. — The RSW estimate (3.8)gives (n) (n) −1 −Cξ | log Cε| (4.2) P L (ψ ) ≤ l ≤ ε ⇒ P L (ψ ) ≤ lC e ≤ Cε 3,3 1,3 (n) Now, if L (ψ ) is less than l,thenboth [0, 1]×[0, 3] and [2, 3]×[0, 3] have a left– 3,3 right crossing of length ≤ l and the restrictions of the ﬁeld to these two rectangles are independent (if r deﬁned in (2.11) is small enough). Consequently, (n) (n) (4.3) P L (ψ ) ≤ l ≤ P L (ψ ) ≤ l 3,3 1,3 (n) Take p small, such that C p < 1 where C is the constant in (4.2) and set r := 0 0 (n) (ψ, p ). (This is not related to r , deﬁned previously.) For i ≥ 0, set 0 0 3,3 (4.4) p := (Cp ) i+1 i (n) (n) −1 (4.5) r := r C exp(−Cξ | log(Cp )|) i+1 i By induction we get, for i ≥ 0, (n) (n) (4.6) P(L (ψ ) ≤ r ) ≤ p 3,3 i Indeed, the case i = 0 follows by deﬁnition and then notice that the RSW estimate (4.2) (n) (n) (n) (n) under the induction hypothesis implies that P(L (ψ ) ≤ r ) ≤ p ⇒ P(L (ψ ) ≤ r ) ≤ 3,3 i 1,3 i+1 (n) (n) (n) (n) 2 2 Cp which gives, using (4.3), P(L (ψ ) ≤ r ) ≤ P(L (ψ ) ≤ r ) ≤ (Cp ) = p . i i i+1 3,3 i+1 1,3 i+1 2 2 −2 From (4.4)weget p = (p C ) C and from (4.5) we have the lower bound, for i 0 i ≥ 1, √ √ i−1 2 i/2 (n) (n) −i −Cξ | log(Cp )| (n) −Ci −Cξ | log p C |2 k 0 k=0 r ≥ (ψ, p )C e ≥ (ψ, p )e e . 0 0 i 3,3 3,3 Our estimate (4.6) then takes the form, for i ≥ 0, 2 i/2 (n) (n) −Ci −ξ C | log p C |2 2 −2 P L (ψ ) ≤ (ψ, p )e e ≤ p C C . 0 0 3,3 3,3 TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 377 This can be rewritten, taking i =2log s,as (n) (n) −1 −Clog s −ξ s −cs P L (ψ ) ≤ (ψ, p )C e e ≤ e 3,3 3,3 for s > 2 with absolute constants. We obtain the statement of the proposition by using again the RSW estimates. Using the comparison result between φ and ψ (Proposition 5), we get the following corollary. Corollary 17 (Lower tail estimates for φ).— For p small enough, but ﬁxed, for all s > 0 we have a constant C < ∞ so that (n) −s −cs (4.7) P L (φ) ≤ e (φ, p) ≤ Ce . 1,3 Upper tails. — The proof for the upper tails is similar to the one of Proposition 5.3 in [18]. Themaindifferenceisthatwehavetoswitchbetween φ and ψ,sothatwecan use the independence properties of ψ together with the scaling properties of φ.Before stating the proposition, we refer the reader to (2.20) for the deﬁnition of (φ, p).In contrast with the lower tails estimates which are relative to (φ, p),wedonot know how to prove (at least a priori) the analogous result for the upper tails with (φ, p) only. However, we can prove it by replacing (φ, p) by (φ, p) (φ, p) and this is the content n n n of the following proposition. Proposition 18 (Upper tail estimates for φ).— For p small enough, but ﬁxed, we have a constant C < ∞ so that for all n ≥ 0 and s > 2, (n) s c log s (4.8) P L (φ) ≥ e (φ, p) (φ, p) ≤ Ce . n n 3,1 Proof. — The proof uses percolation and scaling arguments. A percolation argu- ment is used to build a crossing of a larger rectangle from smaller annular circuits, and then a scaling argument is used to relate quantiles of these annular crossings to crossing quantiles of the larger rectangle. Step 1: Percolation argument. To each unit square P of Z , we associate the four crossings of long rectangles of size (3, 1) surrounding P, each comprising three squares on one side of the eight-square annulus surrounding P, as illustrated in Figure 2.Wedeﬁne (n) S (P,ψ) to be the sum of the four crossing lengths, and declare the site P to be open (n) (n) when the event {S (ψ, P) ≤ 4 (ψ, p)} occurs. This occurs with probability at least 1 − 3,1 (n) (n) ε(p),where ε(p) goes to zero as p goes to zero (recall that P(L (ψ ) ≤ (p)) = 1 − p). 3,1 3,1 Using a highly supercritical ﬁnite-range site percolation estimate to obtain exponential decay of the probability of a left–right crossing (which is standard technique in classical 378 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET FIG. 2. — Four blue rectangles are surrounding the square P. Left–right geodesics associated to the long and short rectangles surrounding P are drawn in green and brown respectively. Any geodesic π , here in red, which intersects P has to cross the green circuit and to induce a short crossing of one of the four rectangles (Color ﬁgure online) percolation theory [20]; see also for example the proof of Proposition 4.2 in [10]) together with the Russo–Seymour–Welsh estimates (to come back to (ψ, p)), we have (n) 2 −ck P L (ψ ) ≥ Ck (ψ, p) ≤ Ce . 3k,k Therefore, using this bound together with Proposition 5 to bound X (recalling the 3k,k deﬁnition (2.18)), (n) ξ C k 2 P L (φ) ≥ e C Ck (φ, p/2) p n 3k,k (n) ξ X ξ C k 2 3k,k ≤ P e L (ψ ) ≥ e C Ck (φ, p/2) p n 3k,k (n) ≤ P X ≥ C k + P L (ψ ) ≥ C Ck (φ, p/2) 3k,k p n 3k,k (n) −ck 2 −ck ≤ Ce + P L (ψ ) ≥ Ck (ψ, p) ≤ Ce . 3k,k ¯ ¯ Note that we used the bound (ψ, p) ≤ C (φ, p/2) from (2.22) in the third inequality; n p n here C is deﬁned as in (2.22). Step 2: Decoupling and scaling. In this step, we give a rough bound of the coarse ﬁeld φ , to obtain spatial independence of the remaining ﬁeld between blocks of size 0,m −m 2 . When an event occurs on one block with high enough probability, the percolation argument of Step 1 then provides, with very high probability, a left–right path of such (n) ξ max φ (m,n) R 0,m 3,1 events occurring simultaneously. Since L (φ) ≤ e L (φ), the scaling prop- 3,1 3,1 (d) (m,n) (n−m) −m erty of the ﬁeld φ, i.e. L (φ) = 2 L m m (φ),gives 3,1 3·2 ,2 (n) ξ s m c 2 P L (φ) ≥ e e (φ, p) n−m 3,1 (n−m) −m c 2 ≤ P max φ ≥ Cm + s m + P 2 L (φ) ≥ e (φ, p) m m 0,m n−m 3·2 ,2 3,1 2 m −cs −c2 ≤ Ce + Ce , TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 379 −1/2 where the ﬁrst term of the second expression is bounded by taking a = C + sm in Proposition 2 and the second bound follows from the result obtained in Step 1 with m Cm k = 2 , taking a slightly larger c in exp(c 2 ) to absorb the factor e . −2ξ k −C k Step 3: We derive an a priori bound (φ, p) ≥ 2 (φ, p)e . (Note that the n n−k −k argument below will be optimized in (5.31).) For each dyadic block of size 2 visited by −k π (φ), one of the four rectangles of size 2 (1, 3) around P has to be crossed by π (φ). n n k −k Therefore, since π (φ) has to visit at least 2 dyadic blocks of size 2 ,wehave (n) k ξ inf φ (k,n) S 2 0,k [0,1] L (φ) ≥ 2 e min min L (R (P), φ), 1,1 i P∈P ,P∩π (φ)=∅ 1≤i≤4 k n S −k where (R (P)) denote the four long rectangles of size 2 (1, 3) surrounding P. Us- 1≤i≤4 ing the supremum tail estimate (2.3) and the left tail estimates (4.7), we get (φ, p) ≥ −2ξ k −C k 2 (φ, p)e . Indeed, n−k ξ inf φ k (k,n) S −2ξ k −C k 2 0,k [0,1] P e min min 2 L (R (P), φ)≤2 (φ, p)e n−k 1≤i≤4 P∈P ,P∩π (φ)=∅ k n ≤ P inf φ ≤−k log 4 − C k 0,k [0,1] k (k,n) S −C k + P min min 2 L (R (P), φ) ≤ (φ, p)e n−k P∈P ,P∩π (φ)=∅ 1≤i≤4 k n and each term is less than p/2 if C is large enough, depending on p. Therefore, we have 2ξ m C m (φ, p) ≤ (φ, p) (φ, p) ≤ 2 e (φ, p) (φ, p). n−m n−m n−m n−m n 2 m Now, by coming back to the partial result obtained in Step 2 and by taking s = 2 for n/2 s ∈[1, 2 ],weget (n) cs log s cs −cs P L (φ) ≥ e e (φ, p) (φ, p) ≤ e . n n 3,1 Step 4: Now we consider large tails, so we assume s ≥ 2 . By a direct comparison −ξ(2n+C n) with the supremum, we have (φ, p) ≥ 2 (lateronwewilluse amoreprecise estimate from [12], see (5.5)). Moreover, bounding from above the left–right distance by taking a straight path from left to right and then using a moment method analogous to (n) ξ s 2(n+1) log 2 the one in (3.10), we get P L (φ) ≥ e ≤ e . Altogether, 1,1 (n) (n) ξ s ξ s P L (φ) ≥ (φ, p) (φ, p)e ≤ P L (φ) ≥ (φ, p)e n n n 1,1 1,1 (s−n log 4−C n) − Cs −c 2(n+1) log 2 log s ≤ e ≤ e e , 380 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET −ξ(2n+C n) where we used (φ, p) ≥ 1 in the ﬁrst inequality and the bound (φ, p) ≥ 2 n n (n) − ξ s 2(n+1) log 2 together with the tail estimate P L (φ) ≥ e ≤ e in the second one. The last 1,1 inequality follow since s ≥ 2 . n/2 Combining the tail estimate of Step 3, valid for s ∈[1, 2 ], and the one of Step 4, n/2 valid for s ≥ 2 , completes the proof. Using again the comparison between φ and ψ given in Proposition 5,weget the following corollary. Corollary 19 (Upper tail estimates for ψ ).— For p small enough, but ﬁxed, we have, for all n ≥ 0 and s > 2, (n) s c log s (4.9) P L (ψ ) ≥ e (ψ, p) (ψ, p) ≤ Ce . n n 3,1 5. Concentration 5.1. Concentration of the log of the left–right crossing length. Condition (T). — Denote by π (ψ ) the left–right geodesic of the unit square associated to the ﬁeld ψ . If there are multiple such geodesics, let π (ψ ) be chosen among them 0,n n in some measurable way, for example by taking the uppermost geodesic. By π (ψ ) its K-coarse graining which we deﬁne as (5.1) π (ψ ) := {P ∈ P : P ∩ π (ψ ) = ∅}, K n recalling the deﬁnition (2.7)of P .Let ψ (P) denote the value of the ﬁeld ψ taken at K 0,n 0,n the center of a block P. We introduce the following condition: there exist constants α> 1, c > 0 so that for K large we have ⎛ ⎛ ⎞ ⎞ 1/α 2ξψ (P) 0,K P∈π (ψ ) ⎜ ⎜ ⎟ ⎟ n −cK (5.2)sup E ≤ e . (Condition (T)) ⎝ ⎝ ⎠ ⎠ n≥K ξψ (P) 0,K P∈π (ψ ) The importance of Condition (T) comes from the following theorem. (n) Theorem 20. — If ξ is such that Condition (T) above is satisﬁed, then (log L (φ) − 1,1 (n) log λ (φ)) is tight, where λ (φ) denotes the median of L . n n≥0 n 1,1 It is not expected that the weight is approximately constant over the crossing (since there may be some large level lines of the ﬁeld that the crossing must cross). Condition (T), however, roughly requires that the length of the crossing is supported by a number of TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 381 coarse blocks that grows at least like some small but positive power of the total number of coarse blocks. Note that the fraction in Condition (T) is the norm of the vector of crossing weights on each block divided by the square of the norm of the same, and thus controlling it amounts to an anticoncentration condition for this vector. The core of this section is the proof of Theorem 20. Before proving it, let us al- ready jump to the important following proposition. Here we use the assumption that ξ ∈ (0, 2/d ), although the formulation of Condition (T) is designed so that it could also hold for larger ξ . Proposition 21. — If γ ∈ (0, 2), then ξ := satisﬁes Condition (T). Proof. — Step 1: Supremum bound. Taking the supremum over all blocks of size −K 2 2 in [0, 1] ,weget 2ξψ (P) 0,K ξ max ψ (P) ξ max φ (P) P∈P 0,K P∈P 0,K K K K e e P∈π (ψ ) n ξ X ≤ ≤ e , ξψ (P) ξψ (P) 0,K 0,K e e K K ξψ (P) P∈π (ψ ) P∈π (ψ ) 0,K n n P∈π (ψ ) recalling the deﬁnition of X below (2.18). Step 2: We give a lower bound of the denominator of the right-hand side. By taking the concatenation of straight paths in each box of π (ψ ), we get a left–right crossing of [0, 1] . Denote this crossing by .Wehave, n,K,ψ ξψ (P) −ξ X ξφ (P) 0,K 1 0,K (5.3) e ≥ e e K K P∈π (ψ ) P∈π (ψ ) n n −ξ X K (K) ≥ e exp(−ξ max osc (φ ))2 L (φ, ) P 0,K n,K,ψ P∈P (K) −ξ X K ≥ e exp(−ξ max osc (φ ))2 L (φ), P 0,K 1,1 P∈P where osc was deﬁned in (2.6)and P was deﬁned in (2.8). Step 3: Combining the two previous steps, we have ξ max φ (P) 1 0,K 2ξψ (P) 0,K P∈P K K e ξ max osc (φ ) P∈π (ψ ) 1 P 0,K n 2ξ X 1 P∈P ≤ e e . 2 (K) 2 L (φ) ξψ (P) 0,K 1,1 P∈π (ψ ) 1 1 Now, we take α> 1 close to 1. Using Hölder’s inequality with + = 1and r close to 1, r s together with Cauchy–Schwarz, we get ⎛ ⎛ ⎞ ⎞ 1/α 2ξψ (P) 0,K P∈π (ψ ) ⎜ ⎜ ⎟ ⎟ ⎝ ⎝ ⎠ ⎠ ξψ (P) 0,K P∈π (ψ ) n 382 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET 1/α αξ max φ (P) P∈P 0,K e αξ max osc (φ ) 1 P 0,K −K 2αξ X 1 P∈P ≤ 2 E e e (K) (L (φ)) 1,1 1/2αs 1/αr −2αs αrξ max φ (P) 1 0,K −K (K) P∈P ≤ 2 E e E L (φ) 1,1 1/4αs 4αsξ max osc (φ ) 1/4αs 1 P 0,K 8αsξ X 1 P∈P × E e E e . Therefore, using (2.4)for themaximum,(4.7) for the left–right crossing, Proposition 5 to bound X and (2.10) for the maximum of oscillations, we ﬁnally get, when αrξ< 2(recall that αr can be taken arbitrarily close to 1), ⎛ ⎛ ⎞ ⎞ α 1/α 2ξψ (P) 0,K ⎜ ⎜ P∈π (ψ ) ⎟ ⎟ (K) n −K 2ξ K −1 C K (5.4) E ≤ 2 2 (φ, p) e . ⎝ ⎝ ⎠ ⎠ 1,1 ξψ (P) 0,K P∈π (ψ ) 2 γ Step 4: Lower bound on quantiles. For γ ∈ (0, 2),Q := + > 2. Using Propo- γ 2 sition 3.17 from [12] (circle average LFPP) and Proposition 3.3 from [11](comparison between φ and circle average), we have, if p is ﬁxed and ε ∈ (0, Q − 2), for K large enough, (K) −K(1−ξ Q+ξε) (5.5) (φ, p) ≥ 2 . 1,1 Step 5: Conclusion. Using the results from the two previous steps, we ﬁnally get ⎛ ⎛ ⎞ ⎞ α 1/α 2ξψ (P) 0,K ⎜ ⎜ P∈π (ψ ) ⎟ ⎟ n −ξ(Q−2−ε)K C K E ≤ 2 e , ⎝ ⎝ ⎠ ⎠ ξψ (P) 0,K P∈π (ψ ) which completes the proof. Now, we come back to the proof of Theorem 20. We ﬁrst derive a priori estimates on the quantile ratios. Lemma 22. — Let Z be a random variable with ﬁnite variance and p ∈ (0, 1/2). If a pair ¯ ¯ ¯ ((Z, p), (Z, p)) satisﬁes (Z, p) ≥ (Z, p), P(Z ≥ (Z, p)) ≥ pand P(Z ≤ (Z, p)) ≥ p, then, we have: (5.6) ((Z, p) − (Z, p)) ≤ Var Z. Proof. — If Z is an independent copy of Z, notice that for l ≥ l we have 2Var(Z) = 2 2 2 E((Z − Z) ) ≥ E(1 1 (Z − Z) ) ≥ P(Z ≥ l )P(Z ≤ l)(l − l) . Z ≥l Z≤l TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 383 (n) In the following lemma, we derive an a priori bound on the variance of log L (φ). 1,1 Lemma 23. — For all n ≥ 0 we have the bound (n) Var log L (φ) ≤ ξ (n + 1) log 2 1,1 (n) Proof. — Denote by L (D ) the left–right distance of [0, 1] for the length metric 1,1 ξφ k −k 0,n e ds,where φ is piecewise constant on each dyadic block of size 2 where it is equal 0,n to the value of φ at the center of this block. (We do not assign an independent meaning 0,n to the notation D .) Note that we have −k −k −C2 ∇φ (n) (n) (n) C2 ∇φ 0,n 0,n 2 2 [0,1] [0,1] e L ≤ L (D ) ≤ L e , 1,1 1,1 1,1 (n) (n) which gives almost surely that L (φ) = lim L (D ). By dominated convergence we k→∞ k 1,1 1,1 have (n) (n) Var log L (φ) = lim Var log L (D ). 1,1 1,1 k→∞ (n) Now, log L (D ) is a ξ -Lipschitz function of p = 4 Gaussian variables denoted by 1,1 Y = (Y , ..., Y ),where on R we use the supremum metric. We can write Y = AN for 1 p some symmetric positive semideﬁnite matrix A and standard Gaussian vector N on R . (n) Then log L (D ) = f (Y) = f (AN) which is ξσ -Lipschitz as a function of N where 1,1 σ = max(|A |, ..., |A |). By the Gaussian concentration inequality of [17, Lemma 2.1], 1 p applied as in [10, Lemma 5.8], since the pointwise variance of the ﬁeld is (n + 1) log 2 we have (n) Var log L (D ) ≤ max(Var(Y ), ..., Var(Y )) = ξ (n + 1) log 2. k 1 p 1,1 Before stating the following lemma, we refer the reader to the deﬁnition of quantile ratios in (2.20). Lemma 24 (A priori bound on the quantile ratios).— Fix p ∈ (0, 1/2). There exists a constant C depending only on p such that for all n ≥ 1, C n (5.7) (ψ, p) ≤ e . (k) Proof. — By using Lemma 23 we get Var(log L (ψ )) ≤ Ck for all 1 ≤ k ≤ n and 1,1 an absolute constant C > 0. This implies the same bound for ψ by Proposition 5. Using (k) (ψ,p) then Lemma 22 with Z = log L (ψ ) for k ≤ n, we ﬁnally get the bound max ≤ k k≤n 1,1 (ψ,p) C n e . 384 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Proof of Theorem 20.— The proof is divided in ﬁve steps. K will denote a large pos- itive number to be ﬁxed at the last step. Step 1. Quantiles-variance relation/setup. We aim to get an inductive bound on (ψ,p/2) (ψ, p). We will therefore bound in term of ’s at lower scales. p will be ﬁxed (ψ,p/2) from now on, small enough so that we have the tail estimates from Section 4 for φ with p and for ψ with p/2. The starting point is the bound (n) (ψ, p/2) C Var log L (ψ ) 1,1 (5.8) ≤ e . (ψ, p/2) Step 2. Efron–Stein. Using the Efron–Stein inequality with the block decomposition of ψ introduced in (2.13), deﬁning the length with respect to the unresampled ﬁeld 0,n (n) L (ψ ) = L (ψ ),weget 1,1 (n) (5.9) Var log L (ψ ) ≤ E log L (ψ ) − log L (ψ ) 1,1 n + E log L (ψ ) − log L (ψ ) , P∈P where in the ﬁrst term (resp. second term) we resample the ﬁeld ψ (resp. ψ )toget 0,K K,n,P ˜ ˜ an independent copy ψ (resp. ψ ) and we consider the left–right distance L (ψ ) 0,K K,n,P (resp. L (ψ )) of the unit square associated to the ﬁeld ψ − ψ + ψ (resp. ψ − 0,n 0,K 0,K 0,n ψ + ψ ). K,n,P K,n,P Step 3. Analysis of the ﬁrst term. For the ﬁrst term, using Gaussian concentration as in the proof of Lemma 23,weget K 2 (5.10) E((log L (ψ ) − log L (ψ )) ) = 2E(Var(log L (ψ )|ψ − ψ )) ≤ CK. n n 0,n 0,K Step 4. Analysis of the second term. For P ∈ P ,if L (ψ ) > L (ψ ), the blockPis K n visited by the geodesic π (ψ ) associated to L (ψ ).Deﬁne n n K ε −K (5.11)P := {Q ∈ P : d(P, Q) ≤ CK 2 }, where we recall that ε is associated with the range of dependence of the resampled ﬁeld ψ through (2.11) (see also the subsection following this deﬁnition). Here, d(P, Q) is K,n,P the L -distance between the sets P and Q. We upper-bound L (ψ ) by taking the concatenation of the part of π (ψ ) outside of P together with four geodesics associated to long crossings in rectangles comprising a K P K circuit around P (for the ﬁeld ψ which coincides with the ﬁeld ψ outside of P ). We 0,n 0,n −K ε K K get, introducing the rectangles (Q (P)) of size 2 (CK , 3) surrounding P (P and i 1≤i≤4 −K its 3 · 2 neighborhood form an annulus, and gluing the four crossings gives a circuit in this annulus) and using the inequality log x ≤ x − 1, P (n) (L (ψ ) − L (ψ )) max L (Q (P), ψ ) n + 1≤i≤4 i (5.12) log L (ψ ) − log L (ψ ) ≤ ≤ 4 . + (n) (n) L (ψ ) L (ψ ) 1,1 1,1 TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 385 FIG. 3. — Illustration of the geodesics used in the upper bound of Step 4 • We recall the notation φ (P) to denote the value of the ﬁeld φ at the center 0,K 0,K of P. We bound from above each term in the maximum of (5.12) as follows: (n) ξ X (n) L (Q (P), ψ ) ≤ e L (Q (P), φ) i i ξ X ξφ (P) ξ osc (φ ) (K,n) 0,K K 0,K ≤ e e e L (Q (P), φ) 2ξ X ξψ (P) ξ osc (φ ) (K,n) 0,K K 0,K ≤ e e e L (Q (P), φ), where the oscillation osc is deﬁned in (2.6)and P is deﬁned in (5.11). −K −K 2 L For a rectangle Q of size 2 , with corners in 2 Z ,wedenoteby (R (Q)) 1≤i≤4 −K the four long rectangles of size 2 (3, 1) surrounding Q. We can upper-bound the rect- angle crossing lengths associated to the Q (P)’s by gluing O(K ) rectangle crossings of −K −K size 2 (3, 1), which include an annulus around each block Q of size 2 (1, 1) (with −K 2 K corners in 2 Z ) in the shaded region A of Figure 3.Weget (K,n) ε (K,n) L max L (Q (P), φ) ≤ CK max L (R (Q), φ) 1≤i≤4 Q∈A ,1≤i≤4 and we end up with the following upper bound: (5.13) log L (ψ ) − log L (ψ ) ξψ (P) 0,K 2ξ X ξ osc (φ ) ε (K,n) L K 0,K 0 ≤ e e CK max L (R (Q), φ). (n) Q∈A ,1≤i≤4 L (ψ ) 1,1 • We lower-bound the denominator of (5.13) as follows. If P ∈ P is visited by a π (ψ ) geodesic, then there are at least two short disjoint rectangle crossings among the four surrounding P. Therefore, if we denote by P the box containing P at its center whose size is three times that of P, 386 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET ξψ (n) S −ξ X (n) S 0,n e ds ≥ 2min L (R (P), ψ ) ≥ e min L (R (P), φ) i i 1≤i≤4 1≤i≤4 π (ψ )∩P −ξ X ξφ (P) −ξ osc (φ ) (K,n) S 0,K 0,K ≥ e e e min L (R (P), φ) 1≤i≤4 −2ξ X ξψ (P) −ξ osc (φ ) (K,n) S 0,K ˆ 0,K ≥ e e e min L (R (P), φ), 1≤i≤4 S −K where (R (P)) denote the four short rectangles of size 2 (1, 3) surrounding P. 1≤i≤4 Summing over all P’s and taking uniform bounds for the rectangle crossings at higher scales, (n) ξψ ξψ 0,n 0,n L (ψ ) = e ds ≥ e ds 1,1 P∩π (ψ ) P∩π (ψ ) n n P∈P P∈P K K −2ξ X (K,n) S ≥ e min min L (R (P), φ) 1≤i≤4 P∈P ⎛ ⎞ ξψ (P) −ξ osc (φ ) 0,K ˆ 0,K ⎝ P ⎠ × e e . P∈P ,P∩π (ψ )=∅ K n Therefore, taking a uniform bound for the oscillation, we get ⎛ ⎞ (n) −2ξ X ξψ (P) −ξ osc (φ ) (K,n) S 0,K 0,K ⎝ ˆ ⎠ (5.14) L (ψ ) ≥ e e e min L (R (P), φ) 1,1 i P∈P ,1≤i≤4 P∈π (ψ ) 1 −ξ max osc (φ ) 1 0,K −2ξ X P∈P P (K,n) S (5.15) ≥ e e min L (R (P), φ) 9 P∈P ,1≤i≤4 ξψ (P) 0,K × e . P∈π (ψ ) L −K • We recall that (R (P)) denote the four rectangles of size 2 (3, 1) surround- 1≤i≤4 ing P. Gathering inequalities (5.13)and (5.15), we have E log L (ψ ) − log L (ψ ) P∈P 2ξψ (P) 0,K (K,n) L e 1 max L (R (P), φ) P∈P ,1≤i≤4 ⎜ P∈π (ψ ) i 2ε n K ≤ CK E 2 S (K,n) min 1 L (R (P), φ) P∈P ,1≤i≤4 ξψ (P) i 0,K P∈π (ψ ) Cξ max osc (φ ) 1 K 0,K ⎟ P 8ξ X P∈P × e e . ⎠ TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 387 • Condition (T) gives us a α> 1and c > 0 so that for K large enough, for n ≥ K, ⎛ ⎛ ⎞ ⎞ 1/α 2ξψ (P) 0,K P∈π (ψ ) ⎜ ⎜ ⎟ ⎟ n −cK E ≤ e . ⎝ ⎝ ⎠ ⎠ ξψ (P) 0,K P∈π (ψ ) Then, by using the gradient estimate (2.10) and recalling the deﬁnition of P in (5.11), we have ε −K +ε Cmax osc (φ ) 1 K 0,K 0 0 CK 2 ∇φ 2 P∈P P 0,K CK [0,1] (5.16) E e ≤ E e ≤ e . It is for the second inequality that in (2.11)wetake ε to be small in the deﬁnition of ψ ; ε < 1/2 is sufﬁcient. Furthermore, using our tail estimates with regard to upper and lower quantiles for φ (see (4.7)and (4.8), and the scaling property (2.23), for β> 1so that 1 1 + = 1, we get α β ⎛ ⎞ 1 2β (K,n) L max 1 L (R (P), φ) +ε P∈P ,1≤i≤4 i 0 K 2 CK ⎝ ⎠ (5.17) E ≤ (φ, p)e . n−K (K,n) min 1 L (R (P), φ) P∈P ,1≤i≤4 Note that we could have a log K term instead of the K in (5.17). Altogether, by applying Hölder inequality and Cauchy–Schwarz, we get +ε 2 0 P −cK CK 2 (5.18) E log L (ψ ) − log L (ψ ) ≤ e e (φ, p) n n−K P∈P +ε −cK CK 2 ≤ e e C (ψ, p/2), n−K 2 2 where we used (2.22) in the last inequality to get (φ, p) ≤ C (ψ, p/2). n−K n−K Step 5. Conclusion. Gathering the bounds obtained in Step 3 (inequality (5.10)) and Step 4 (inequality (5.18)), we get, coming back to the inequality (5.9), for K large enough, (n) −C K 2 (5.19)VarlogL (ψ ) ≤ C K + e (ψ, p/2). 1,1 n−K Now, we will show that this bound together with the a priori bound on the quantile ratios (Lemma 24) is enough to conclude ﬁrst that (ψ, p/2)< ∞ and then that (n) sup Var log L (ψ ) < ∞, using the tail estimates (4.7)and (4.9). n≥0 1,1 Coming back to Step 1 (equation (5.8)) and using (5.19), we get the inductive in- equality (5.20) below for K large enough and n ≥ K, and (5.21) below by the a priori bound on the quantile ratios Lemma 24: (n) (ψ, p/2) 2 −C K n 2 C Var log L (ψ ) C C K+e (ψ,p/2) 1,1 p 1 n−K (5.20) ≤ e ≤ e ; (ψ, p/2) C K (5.21) (ψ, p/2) ≤ e . K 388 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET From now on, we take K large enough but ﬁxed so that √ √ −C K C K C 2C K 2 2 p p 1 (5.22) e (e + e ) ≤ C K. Set C 2C K p 1 (5.23) := (ψ, p/2) ∨ e , Rec K so that (ψ, p/2) ≤ . This is the initialization of the induction. Now, assume that K Rec (ψ, p/2) ≤ . In particular, (ψ, p/2) ≤ and using (5.20) n−1 Rec n−K Rec (ψ, p/2) −C K 2 C C K+e p 1 Rec ≤ e (ψ, p/2) C 2C K p 1 The right-hand side is smaller than e and therefore than . Indeed, by (5.23), Rec (5.21)and (5.22), −C K 2 −C K C 2C K 2 2 2 p 1 e ≤ e ( (ψ, p/2) + e ) Rec −C K C K C 2C K 2 2 p p 1 ≤ e (e + e ) ≤ C K. Therefore, (ψ, p/2) (ψ, p/2) = (ψ, p/2) ∨ ≤ . n n−1 Rec (ψ, p/2) Therefore, (ψ, p/2)< ∞ thus (φ, p)< ∞ and by the tail estimates (4.7)and (4.8), ∞ ∞ (n) the sequence (log L (φ) − log λ (φ)) is tight. n n≥0 1,1 5.2. Weak multiplicativity of the characteristic length and error bounds. — Henceforth, we will only consider the case ξ = for γ ∈ (0, 2) and the ﬁeld φ . All observables will 0,n be assumed to be taken with respect to φ and we will drop the additional notation used to differ between φ and ψ . In this case, we saw that there exists a ﬁxed constant C > 0 (n) (n) (n) (n) −1 ¯ ¯ so that for all n ≥ 0, (p) ≤ C (p),C (p) ≤ (p) and with the tail estimates, 3,1 1,3 3,1 1,3 (n) (n) E(L ) ≤ CE(L ). All these characteristic lengths are uniformly comparable. We will 3,1 1,3 (n) take λ to denote one of them, say the median of L . 1,1 In the next elementary lemma, we prove that a sequence satisfying a certain quan- titative weak multiplicative property has an exponent, and we quantify the error. Lemma 25. — Consider a sequence of positive real numbers (λ ) . If there exists C > 0 such n n≥1 that for all n ≥ 1,k ≥ 1 we have √ √ −C k C k (5.24) e λ λ ≤ λ ≤ e λ λ , n k n+k n k n+O( n) then there exists ρ> 0 such that λ = ρ . n TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 389 Proof. — We introduce the sequence (a ) such that λ n+1 = (λ n ) e . By iterat- n n≥0 2 2 ing, we get n+1 n n−1 2 a 4 2a +a 2 2 a +2 a +···+2a +a n n−1 n 0 1 n−1 n n+1 n n−1 λ = (λ ) e = (λ ) e = ··· = λ e . 2 2 2 −n/2 The condition (5.24) gives that the sequence 2 a is bounded, therefore the series n≥0 a a −k/2 −n/2 k k converges and | |≤ 2 (sup 2 |a |) 2 . In particular there exists k k k k≥0 k≥n k≥0 2 2 ρ> 0such that a a a 1 n 1 ∞ n+1 k n+1 k n k n n/2 2 log λ + 2 log λ + −2 1 1 2 O(2 ) 2 k=0 k 2 k=0 k k≥n+1 k 2 2 2 n+1 λ = e = e e = ρ e . Now that we have the existence of an exponent, we prove the upper bound of Lemma 25. There exist C , C > 0 such that we have the following upper bounds: 1 2 k k/2 2 C 2 (5.25) λ k ≤ ρ e , C k (5.26) λ ≤ λ λ e . n+k n k 2 2 C Take C large enough so that (C + C ) + (C + C )C ≤ C and λ ≤ ρe .Wewantto 3 1 2 1 2 3 1 n C n prove by induction that for all n ≥ 1, λ ≤ ρ e . The assumption on C implies that this n 3 k k+1 holds for n = 1. By induction (in a dyadic fashion), take n ∈[2 , 2 ). We decompose n as k k n = 2 + n with n ∈[0, 2 ). We have, by using (5.26), (5.25) and the induction hypothesis, k k k/2 k k/2 k/2 C 2 2 C 2 n C n C 2 2 1 k 3 k 2 λ ≤ λ k λ e ≤ (ρ e )(ρ e )e n 2 n √ √ k/2 n (C +C )2 +C n n C n 1 2 3 k 3 = ρ e ≤ ρ e , since by the assumption on C we have √ √ k/2 2 k k/2 2 (C + C )2 + C n = (C + C ) 2 + (C + C )C 2 n + C n 1 2 3 k 1 2 1 2 3 k k 2 k 2 ≤ C (2 + n ) = C n. 3 3 The proof of the lower bound is similar. In the next proposition we prove that the characteristic length λ satisﬁes the weak multiplicativity property (5.24) and we identify the exponent by using the results of [12]. Proposition 26. — For ξ satisfying Condition (T), there exists C > 0 such that for all n ≥ 1, k ≥ 1 we have √ √ −C k C k (5.27) e λ λ ≤ λ ≤ e λ λ . n k n+k n k Furthermore, when γ ∈ (0, 2) and ξ = γ/d , we have −n(1−ξ Q)+O( n) (5.28) λ = 2 . n 390 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Proof. — Let us assume ﬁrst that (5.27) holds. Then, by using Lemma 25,there n+O( n) exists ρ> 0such that we have λ = ρ . Similarly to (5.5), for each ﬁxed small δ> 0, for k large enough we have, −k(1−ξ Q−δ) (5.29) λ ≤ 2 . The proof of (5.29)follows thesamelinesasthe oneof(5.5). Combining (5.29)and (5.5) −(1−ξ Q) we get ρ = 2 . Now, we prove that the characteristic length satisﬁes (5.27). (k) (k) Step 1: Weak submultiplicativity. Let π be such that L (π ) = L .If P ∈ P is vis- k k k 1,1 (k,n+k) ξφ k,n+k ited by π , consider the concatenation S (P) of four geodesics for e ds associated −k to the rectangles of size 2 (3, 1) surrounding P. Each geodesic is in the long direction (k,n+k) (k,n+k) of its rectangle so that this concatenation is a circuit. By scaling, E(L (S (P))) = (n) −k+2 k 2 E(L ). Note that the collection π (φ) ={P ∈ P : P ∩ π = ∅} is measurable with k k 3,1 k (k,n+k) respect to φ , which is independent of φ .Set := k S (P).Notethat 0,k k,n+k k,n P∈π (φ) contains a left–right crossing of [0, 1] whoselengthisbounded aboveby k,n (n+k) (n+k) (k,n+k) L ( ) = L (S (P)) k,n P∈π (φ) (k,n+k) (k,n+k) ξφ (P) ξ osc (φ ) 0,k 0,k ≤ L (S (P))e e , P∈π (φ) where P denotes the box containing P at its center whose side length is three times that (n+k) (n+k) of P. Since L ≤ L ( ), by independence we have k,n 1,1 ⎛ ⎞ (n+k) (n) −k ξφ (P) ξ osc (φ ) 0,k ˆ 0,k ⎝ P ⎠ E(L ) ≤ 4E(L )E 2 e e . 1,1 3,1 P∈π (φ) −k If P is visited, then one of the four rectangles of size 2 (1, 3) in P surrounding P contains a short crossing, denoted by π ˜ (P) and we have ξφ (k) −k ξ inf φ −k ξφ (P) −ξ osc (φ ) 0,k ˆ 0,k 0,k ˆ 0,k P P e 1 ds ≥ L (π ˜ (P)) ≥ 2 e ≥ 2 e e , ˆ k π ∩P hence −k ξφ (P) ξ osc (φ ) 2ξ osc (φ ) ξφ 0,k ˆ 0,k ˆ 0,k 0,k P P 2 e e ≤ e e 1 ds. π ∩P k k P∈π (φ) P∈π (φ) k k Taking the supremum of the oscillation over all blocks, −k 2ξ 2 ∇φ (k) 2ξ osc (φ ) ξφ 0,k 0,k 0,k 2 ˆ [0,1] e e 1 ds ≤ 9e L . π ∩P 1,1 P∈π (φ) k TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 391 Altogether, by Cauchy–Schwarz, −k (n+k) (n) (k) 4ξ 2 ∇φ 2 1/2 0,k 1/2 [0,1] E(L ) ≤ 36E(L )E((L ) ) E(e ) . 1,1 3,1 1,1 When ξ satisﬁes Condition (T), by using the uniform bounds for quantile ratios together C k with the upper tail estimates (4.8) and the gradient estimate (2.10)weget λ ≤ e λ λ . n+k n k Step 2: Weak supermultiplicativity. We argue here that −C k (5.30) λ ≥ e λ λ . n+k n k Using a slightly easier argument than (5.15) (since we just have the ﬁeld φ here), we have −ξ max osc (φ ) 1 0,k (n+k) ˆ P∈P P (k,k+n) S ξφ (P) 0,k L ≥ e min L (R (P)) e , 1,1 i P∈P ,1≤i≤4 P∈π n+k where π denotes the k-coarse grained approximation of π , the left–right geodesic of n+k n+k 2 S [0, 1] for the ﬁeld φ ,and wherewerecallthat (R (P)) denote the four rectan- 0,n+k 1≤i≤4 −k gles of size 2 (1, 3) surrounding P. Furthermore, by using a similar argument to (5.3), we have (k) ξφ (P) −ξ max osc (φ ) k 0,k P∈P P 0,k e ≥ e 2 L . 1,1 P∈π n+k Altogether, we get the following weak supermultiplicativity, (n+k) (k) k (k,k+n) S −2ξ max osc (φ ) P∈P P 0,k (5.31)L ≥ L min 2 L (R (P)) e 1,1 1,1 P∈P ,1≤i≤4 When ξ satisﬁes Condition (T), by scaling and the tail estimates (4.7), k (k,k+n) S −C k −ck P(min 1 2 L (R (P)) ≥ λ e ) ≥ 1 − e . Furthermore, using the gradient P∈P ,1≤i≤4 n −k −ck estimates (2.9), we get P(2 ∇φ ≥ C k) ≥ 1 − e for C large enough. There- 0,k 2 [0,1] √ √ (n) −C k −C k fore, with probability ≥ 1/2, L ≤ e λ λ hence the bound λ ≥ e λ λ . n k n+k k n 1,1 5.3. Tightness of the log of the diameter. 2 −1 ξφ 0,n Proposition 27. — If γ ∈ (0, 2) and ξ = γ/d then log Diam [0, 1] ,λ e ds n≥0 is tight. Proof. — Step 1: Chaining. By a standard chaining argument, (see (6.1) in [18]for more details), we have 2 ξφ 2 (n) −n ξ sup φ 0,n 2 0,n [0,1] (5.32)Diam [0, 1] , e ds ≤ C max L (P) + C × 2 e , P∈C k=0 k −k where C is a collection of no more than C4 long rectangles of side length 2 (3, 1). k 392 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET ξ sup φ −n 2 0,n [0,1] Using the bound for the maximum (2.4), when ξ< 2, we have E(2 e ) ≤ −n 2ξ n C n 2 2 e . (n) Fix 0 ≤ k ≤ n and P ∈ C . We can bound L (P) by taking a left–right geodesic π k k,n for φ . Therefore, k,n (n) (n) ξ max φ (k,n) 2 0,k [0,1] L (P) ≤ L (π ) ≤ e L (P), k,n and consequently, (n) ξ max φ (k,n) 2 0,k [0,1] (5.33)max L (P) ≤ e max L (P). P∈C P∈C k k Using independence, the maximum bound (2.4), scaling of the ﬁeld φ and the tail estimates (4.8), we get +ε ξ max φ 2 (k,n) −k 2ξ k C k Ck 2 0,k [0,1] (5.34) E e max L (P) ≤ 2 2 e λ e n−k P∈C forsomeﬁxedsmall ε> 0 (again, the term k could in fact be log k). Taking the expec- tation in (5.32), using (5.33)and (5.34), we obtain the following bound for the expected value of the diameter, +ε 2 ξφ −k 2ξ k Ck 0,n (5.35) E(Diam([0, 1] , e ds)) ≤ C 2 2 λ e . n−k k=0 C k k(1−ξ Q) C k Step 2: Right tail. By Proposition 26, λ ≤ λ ≤ λ 2 e . Together with n−k n n (5.35), this implies that +ε 2 ξφ −k 2ξ k Ck 0,n E(Diam([0, 1] , e ds)) ≤ C 2 2 λ e n−k k=0 +ε −kξ(Q−2) Ck ≤ λ C 2 e . k=0 2 −1 ξφ s −s 0,n Since Q > 2, Markov’s inequality gives P Diam([0, 1] ,λ e ds) ≥ e ≤ Ce . Step 3: Left tail. Finally, since the diameter of the square [0, 1] is larger than the 2 −1 ξφ −s 0,n left–right distance, by our tail estimates (4.7), we get P Diam([0, 1] ,λ e ds) ≤ e ≤ (n) −s −cs P L ≤ λ e ≤ Ce . 1,1 TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 393 5.4. Tightness of the metrics. −1 ξφ 0,n Proposition 28. — If γ ∈ (0, 2) and ξ = γ/d then the sequence of metrics λ e ds n≥0 is tight. Moreover, if we deﬁne |x − x | d (x, x ) 0,n n n C := sup and C := sup α β 2 d (x, x ) 2 |x − x | 0,n x,x ∈[0,1] x,x ∈[0,1] n n then, for α> ξ(Q + 2) and β< ξ(Q − 2), the sequence (C , C ) is tight. n≥0 α β −1 ξφ 0,n Henceforth, we use the notation d for the renormalized metric λ e ds re- 0,n stricted to [0, 1] . Proof. — Theproof hastwo parts. In theﬁrstpartweshowthe tightness of the 2 2 + metrics in the space of continuous function from [0, 1] ×[0, 1] → R and in the second part we show that subsequential limits are metrics. A byproduct result of the argument is explicit bi-Hölder bounds. Part 1. Upper bound on the modulus of continuity. We suppose γ ∈ (0, 2).We start by proving that for every 0 <β <ξ(Q − 2),if ε> 0, there exists a large C > 0so that for every n ≥ 0 2 β (5.36) P ∃x, x ∈[0, 1] : d (x, x ) ≥ C |x − x | ≤ ε, 0,n ε β 2 i.e. d is tight, where the C -norm is deﬁned for f :[0, 1] × 0,n β 2 2 C ([0,1] ×[0,1] ) n≥0 [0, 1] → R as β 2 2 C ([0,1] ×[0,1] ) |f (x, y) − f (x , y )| := f + sup . 2 2 [0,1] ×[0,1] |(x, y) − (x , y )| 2 2 (x,y)=(x ,y )∈[0,1] ×[0,1] −n s By a union bound it sufﬁces to estimate P(∃x, x :|x − x | < 2 , d (x, x ) ≥ e |x − 0,n x | ) and −k −k+1 s β P ∃x, x : 2 ≤|x − x |≤ 2 , d (x, x ) ≥ e |x − x | . 0,n k=0 −k −k+1 s Step 1: We start with the term P(∃x, x : 2 ≤|x − x |≤ 2 , d (x, x ) ≥ e |x − 0,n x | ). We use the chaining argument (5.32)atscale k which gives ξ sup φ 0,n −1 (n) −1 −n 2 [0,1] sup d (x, x ) ≤ Cλ max L (P) + Cλ × 2 e . 0,n n n P∈C −k −k+1 i 2 ≤|x−x |≤2 i=k 394 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Taking the expected value and using the same bounds as those obtained in the proof of Proposition 27,weget 1 1 +ε +ε 2 2 −iξ(Q−2) Ci −kξ(Q−2) Ck E sup d (x, x ) ≤ 2 e ≤ C2 e . 0,n −k −k+1 2 ≤|x−x |≤2 i=k Therefore, using Markov’s inequality we get the bound −k −k+1 s β P ∃x, x : 2 ≤|x − x |≤ 2 , d (x, x ) ≥ e |x − x | 0,n k=0 n n s −kβ −s kβ −kξ(Q−2) ≤ P sup d (x, x ) ≥ e 2 ≤ e 2 2 . 0,n −k −k+1 2 ≤|x−x |≤2 k=0 k=0 The series is convergent since ξ(Q − 2) − β> 0. −n s β Step 2: We bound from above P(∃x, x |x − x | < 2 , d (x, x ) ≥ e |x − x | ) using 0,n a bound on the supremum of the ﬁeld. Indeed, for such x and x ,notethat ξ sup φ s β −1 2 0,n [0,1] e |x − x | ≤ d (x, x ) ≤ λ e |x − x | 0,n Writing β = ξ(Q − 2) − εξ for some ε> 0, it follows that 1 − β = (1 − ξ Q + 2ξ) + εξ > 0 since the LFPP exponent 1 − ξ Q ≥−2ξ by a simple uniform bound. Therefore, β−1 n(1−β) −1 n(1−β) n(2ξ +εξ +o(1)) |x − x | ≥ 2 and λ 2 = 2 . Altogether, this probability is bounded −1 from above by P(sup φ ≥ n log 4 + εn log 2 + o(n) + ξ s) and using (2.3)gives a 2 0,n [0,1] uniform tail estimate. Therefore, we obtain the tightness of d as a random element of C([0, 1] × 0,n n≥0 2 + [0, 1] , R ) and every subsequential limit is (by Skorohod’s representation theorem) a pseudo-metric. Part 2. Lower bound on the modulus of continuity. We prove that if α> ξ(Q + 2) and ε> 0 then there exists a small constant c > 0such that for every n ≥ 0, 2 α (5.37) P ∃x, x ∈[0, 1] : d (x, x ) ≤ c |x − x | ≤ ε. 0,n ε Similarly as before, by union bound it is enough to estimate the term 2 −n −ξ s α (5.38) P(∃x, x ∈[0, 1] :|x − x | < 2 , d (x, x ) ≤ e |x − x | ) 0,n and the term ⎛ ⎞ ⎜ ⎟ −k −k+1 −ξ s α (5.39) P ∃x, x : 2 ≤|x − x |≤ 2 , d (x, x ) ≤ e |x − x | . ⎝ 0,n ⎠ k=0 :=E k,n,s TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 395 2 −k Step 1: We give an upper bound for (5.39). Fix x, x ∈[0, 1] such that 2 ≤|x − −k+1 x |≤ 2 . Note that any path from x to x crosses one of the rectangles in the collection S 1 {R (P) : P ∈ P , 1 ≤ i ≤ 4}. Hence, under the event E , there exists x, x such that k,n,s i k+2 −kα −1 −k ξ inf φ k (k,n) S 2 0,k [0,1] (5.40)2 ≥ d (x, x ) ≥ λ 2 e min 2 L (R (P)) . 0,n n i P∈P ,1≤i≤4 k+2 Since α = ξ(Q + 2) + ξδ for a small δ> 0, by using Proposition 26 we get √ √ √ −kα k −kα C k −k(α−ξ Q) C k −k(2+δ)ξ −ξδk C k (5.41)2 λ 2 ≤ 2 λ λ e ≤ 2 λ e = 2 (λ 2 e ) n k n−k n−k n−k Now, using (5.40), (5.41) and scaling, we get ξ inf φ k (k,n) S −kα k −ξ s 2 0,k [0,1] P E ≤ P e min 2 L (R (P)) ≤ 2 λ 2 e k,n,s n P∈P ,1≤i≤4 k+2 ≤ P sup |φ |≥ k log 4 + kδ log 2 + s/2 0,k [0,1] (n−k) S −kδξ C k −ξ s/2 + P min L (R (P)) ≤ λ 2 e e n−k P∈P ,1≤i≤4 k+2 −ck −cs ≤ Ce e , where we used in the last inequality the supremum bounds (2.3) and the left tail estimate (4.7). Step 2: Finally, we control (5.38). We write −n −ξ s α P(∃x, x :|x − x | < 2 , d (x, x ) ≤ e |x − x | ) 0,n d (x, x ) 0,n −ξ s ≤ P inf ≤ e −n |x−x |≤2 |x − x | −1 ξ inf φ 1−α −ξ s 2 0,n [0,1] ≤ P λ e inf |x − x | ≤ e . −n |x−x |≤2 We recall that α> ξ Q + 2ξ , and in particular α> 1: indeed, 1 − ξ Q ≤ 2ξ follows from 1−α −n(1−α) a comparison with the inﬁmum of the ﬁeld. In this case, inf −n |x − x | = 2 , |x−x |≤2 and by Proposition 26, √ √ −n(1−α) −1 −n(1−α) n(1−ξ Q) −C n n(α−ξ Q) −C n 2 λ ≥ 2 2 e = 2 e n 396 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Therefore, since α − ξ Q = 2ξ + δξ for some δ> 0, we have for n large that ξ inf φ −1 1−α −ξ s 2 0,n [0,1] P λ e inf |x − x | ≤ e −n |x−x |≤2 ≤ P sup |φ |≥ n log 4 + n log 2 + s 0,n [0,1] Using (2.3) completes the proof. Acknowledgements We would like to thank the referees for many helpful comments which helped to improve the exposition of the manuscript. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. Appendix A A.1 Comparison with the GFF molliﬁed by the heat kernel Let h be a GFF with Dirichlet boundary condition on a domain D and U ⊂⊂ Dbe a subdomain of D. We recall that we denote by p the two-dimensional heat kernel at time t |x| 1 − 2t i.e. p (x) = e . The goal of this section is to obtain a uniform estimate to conclude on 2π t the tightness of the renormalized metric associated to p ∗ h assuming the one associated to φ . In particular, the second assertion of Theorem 1 is a corollary of the following proposition. Proposition 29. — There exist constants C, c > 0 such that for all t ∈ (0, 1/2), there is a (d) coupling of h and ϕ = φ such that for all x ≥ 0, we have −cx P ϕ − p ∗ h ≥ x ≤ Ce . U TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 397 Molliﬁcation of the GFF by the heat kernel. — The covariance of the Gaussian ﬁeld p ∗ h is given for x, x ∈ Uby t t t t E p ∗ h(x) p ∗ h(x ) = p (x − y)G (y, y )p (y − x )dydy , 2 2 2 2 D D where G is the Green function associated to the Laplacian operator on D. For an open set A, we denote by p (x, y) the transition probability density of a Brownian motion killed upon exiting A. White noise representation. — Take a space–time white noise W and deﬁne the ﬁeld η on Uby (A.1) η (x) := p ∗ p s (x, y)W(dy, ds) 0 D D D t t where p ∗ p s (x, y) := p (x − y )p s (y , y)dy , 2 2 2 2 (d) so that (η (x)) = (p t ∗ h(x)) . Indeed, by Fubini, we have t x∈U x∈U E(η (x)η (x )) t t D D t t = p ∗ p (x, y) p ∗ p (x , y)dyds s s 2 2 2 2 0 D D D t t = p (x − y )p s (y , y) p (x − y ) p s (y , y)dydy dy ds 2 2 2 2 0 D D D D D t t = p (x − y ) p s (y , y)p s (y, y )dyds p (x − y )dy dy 2 2 2 2 D D 0 D t t = p (x − y )G (y , y )p (y − x )dy dy . 2 2 D D 1 (d) Coupling. — Note that for t ∈ (0, 1/2)φ (x) = p (x − y)W(dy, ds) = ϕ (x),where 2 t t R 2 we set 1−t t+s ϕ (x) := p (x − y)W(dy, ds). 0 R 1 2 Furthermore, we can decompose ϕ (x) = ϕ (x) + ϕ (x),where t t 1−t t+s (A.2) ϕ (x) := p (x − y)W(dy, ds); 0 D 1−t t+s (A.3) ϕ (x) := p (x − y)W(dy, ds). 0 D 398 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET 1 2 Recalling the deﬁnition of η in (A.1), we introduce η and η so that t t 1−t ∞ D D t t (A.4) η (x) = p ∗ p s (x, y)W(dy, ds) + p ∗ p s (x, y)W(dy, ds) 2 2 2 2 0 D 1−t D 1 2 =: η (x) + η (x). t t Therefore, under this coupling (viz. using the same white noise W), we have 1−t 1 1 D (A.5) ϕ (x) − η (x) = p t+s (x − y) − p ∗ p (x, y) W(dy, ds). t t 2 2 0 D Comparison between kernels. — We will consider x, y ∈ U, subdomain of D. Set d := d(U, D )> 0. D D D t t t p ∗ p s (x, y) := p (x − y )p s (y , y)dy = p (x − y )p (y − y)q s (y , y)dy , 2 2 2 2 2 2 D D where q (x, x ) is the probability that a Brownian bridge between x and x with lifetime t stays in D. Therefore, using Chapman–Kolmogorov, t t+s t s p ∗ p s (x, y) − p (x, y) =− p (x − y )p (y − y)dy 2 2 2 t s + p (x − y )p (y − y)(q s (y , y) − 1)dy . 2 2 Note that the ﬁrst term can be bounded by using that |y − y |≥ d for y ∈ Uand y ∈ D . For the second term, we can split the integral over D in two parts: one over the ε- neighborhood of ∂ D (within D), denoted by (∂ D) , and one over its complement. To give ε ε an upper bound on the ﬁrst, we use that for y ∈ Uand y ∈ (∂ D) , |y − y |≥ d(U,(∂ D) ). Finally, we bound the second part by using a uniform estimate on the probability that a Brownian bridge between a point in U and a point D \ (∂ D) exits D in time less than s/2. (Note that 1 − q s (y, y ) is the probability that a Brownian bridge between y and y with time length s/2 exits D.) Therefore, we get that uniformly in x, y ∈ Uand t, D − (A.6) |p ∗ p (x, y) − p t+s (x, y)|≤ Ce . 2 2 Comparison between ϕ and p ∗ h. — By the triangle inequality, 1 1 2 2 (A.7) ϕ − p ∗ h ≤ ϕ − η + ϕ + η . t t t t U U U We look for a uniform right tail estimate (in t) of each term in the right-hand side of (A.7). In order to do so, we will use the Kolmogorov continuity criterion. Therefore, we derive below some pointwise and difference estimates. TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 399 First term. We derive ﬁrst a pointwise estimate. For x ∈ U, using the kernel comparison (A.6), there exists some C > 0 such that, uniformly in t, 1−t 1 2 D t t+s Var η (x) − ϕ (x) = p ∗ p s (x, y) − p (x, y) dyds t t 2 2 0 D 1−t ≤ C e ds ≤ C . 1 1 We now give a difference estimate: introducing (x) := ϕ (x) − η (x),for x, x ∈ U, t t 1−t E (x) − x = p t+s (x − y) − p t ∗ p (x, y) t t 2 2 0 D t+s t − p x − y − p ∗ p s x , y dyds, 2 2 which is uniformly bounded in t ∈ (0, 1/2) by a quantity of size O(|x − x |). (By splitting the integral at |x − x |,one canuse (A.6) for the small values of s and gradient estimates for both kernels for larger values of s.) Second term. We recall here that ϕ (x) is deﬁned for x ∈ Uby 1−t 1 (d) t+s ϕ (x) = p (x − y)W(dy, ds) = p (x − y)W(dy, ds). 2 2 c c 0 D t D We have, for x, x ∈ U, with d := d(U, D ), 2 2 E ϕ (x) − ϕ (x ) t t s s ≤ p (x − y) − p (x − y) dyds 2 2 t D s s ≤ p (x − y) − p (x − y) dyds 2 2 |x−x | R |x−x | s s + p (x − y) − p (x − y) dyds 2 2 0 D 1 |x−x | ≤ 2 p (0) − p (x − x ) ds + 4 p (d)ds ≤ C|x − x |, s s |x−x | 0 −z whereweuse 1 − e ≤ z in the last inequality. Similarly, we can prove that there exists C > 0 independent of t such that E(φ (x) ) ≤ C. 2 2 Third term. We recall here that η (x) is deﬁned for x ∈ Uby η (x) = p t ∗ t t 1−t D p s (x, y)W(dy, ds). Similarly, there exists C > 0such that for t ∈ (0, 1/2), x, x ∈ U, we 2 400 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET have 2 2 D D t t E η (x) − η (x ) ≤ p ∗ p s (x, y) − p ∗ p s (x , y) dyds t t 2 2 2 2 1/2 D ≤ C|x − x |. Furthermore, the pointwise variance is uniformly bounded. Result. Altogether, coming back to (A.7) and combining Kolmogorov continuity criterion with Fernique’s theorem (see Section 1.3 in [22]), we get the following tail esti- mate on the above coupling: there exist C, c > 0such that for all t ∈ (0, 1/2), x ≥ 0, we have −cx P ϕ − p ∗ h ≥ x ≤ Ce . A.2 Approximations for δ ∈ (0, 1) −n We explain here how results obtained along the sequence {2 : n ≥ 0} can be extended −(n+r) to δ ∈ (0, 1).For each δ ∈ (0, 1),let n ≥ 0and r ∈[0, 1] such that δ = 2 .Thenby decoupling the ﬁeld φ , using a uniform estimate for r ∈[0, 1] and a scaling argument, 0,r −n we generalize our previous results obtained along the sequence 2 to δ ∈ (0, 1). Decoupling low frequency noise. — Note that there exists C > 0such that for n ≥ 0and r ∈[0, 1] we have −C C (A.8) e λ ≤ λ ≤ λ e . n n+r n (r,n+r) (n+r) (r,n+r) −ξ inf φ ξ sup φ 2 0,r 2 0,r [0,1] [0,1] Indeed, note that a.s. e L ≤ L ≤ e L . Furthermore, with 1,1 1,1 1,1 (d) (r,n+r) (n) −r high probability sup |φ |≤ C ≤ C. Then, note that L = 2 L and a.s. 2 r r 0,r r [0,1] 1,1 2 ,2 (n) (n) (n) L ≤ L ≤ L . By the tightness result, there exists a constant C > 0 such that uni- r r 1,2 2 ,2 2,1 (n) (n) −C C formly in n, with high probability, L ≥ e λ and L ≤ e λ , therefore, with high n n 1,2 2,1 (r,n+r) −C C probability, e λ ≤ L ≤ e λ ,hence (A.8). n n 1,1 Weak multiplicativity. — In this paragraph, we will use the notation λ from the introduc- −n tion. We recall that writing λ instead of λ was an abuse of notation. Now we prove n 2 that there exists C > 0such that for δ, δ ∈ (0, 1) we have √ √ −1 −C | log δ∨δ | C | log δ∨δ | (A.9)C e λ λ ≤ λ ≤ Ce λ λ . δ δ δδ δ δ Similarly as (A.8), there exists C > 0such that for r, r ∈[0, 1], n, n ≥ 0, −C C (A.10) e λ ≤ λ ≤ λ e . −n−n −n−r−n −r −n−n 2 2 2 TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 401 −(n+r) −(n +r ) For δ, δ ∈ (0, 1),let n, n ≥ 0and r, r ∈[0, 1] such that δ = 2 , δ = 2 .Note that n =[− log δ]. Using the weak multiplicativity for powers of 2, we have √ √ −C n∧n C n∧n −n −n (A.11) e λ λ −n ≤ λ −n−n ≤ λ λ −n e . 2 2 2 2 2 Without loss of generality, we consider just the upper bound in (A.9). The lower bound follows along the same lines. By using ﬁrst (A.10)and then (A.11)weget C C n∧n C λ = λ ≤ λ e ≤ λ −n λ e e . −n−r−n −r −n−n −n δδ 2 2 2 2 Now, the result follows by using (A.8): √ √ C n∧n C n+r∧n +r 2C C log |δ∨δ | 2C −n −n−r λ λ e ≤ λ λ e e = λ λ e e . −n −n −r 2 2 δ δ 2 2 Tail estimates and tightness of metrics. — Using the same argument as in the two previous −n paragraphs and the tail estimates obtained along the sequence {2 : n ≥ 1},wehavethe following tail estimates for crossing lengths of the rectangles [0, a]×[0, b]: there exists c, C > 0 (depending only on a, b and γ )suchthatfor s > 2, uniformly in δ ∈ (0, 1),we have −1 (δ) s −c log s (A.12) P λ L ≥ e ≤ Ce ; δ a,b −1 (δ) −s −cs (A.13) P λ L ≤ e ≤ Ce . δ a,b −1 ξφ 2 Furthermore, the sequence of metrics (λ e ds) on [0, 1] is tight. δ∈(0,1) REFERENCES 1. M. ANG, Comparison of discrete and continuum Liouville ﬁrst passage percolation, Electron. Commun. Probab., 24 (2019), 2. J. ARU,E.POWELL and A. 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Department of Mathematics, Stanford University, Stanford, CA 94305, USA Current address: Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA H. F. Department of Mathematics, Columbia University, New York, NY 10027, USA Manuscrit reçu le 20 novembre 2019 Version révisée le 23 octobre 2020 Manuscrit accepté le 30 octobre 2020 publié en ligne le 4 novembre 2020. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals http://www.deepdyve.com/lp/springer-journals/tightness-of-liouville-first-passage-percolation-for-0-2-documentclass-7kHyrjHY5S

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Publisher: Springer Journals
Copyright: Copyright © IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2020
ISSN: 0073-8301
eISSN: 1618-1913
DOI: 10.1007/s10240-020-00121-1
Publisher site: See Article on Publisher Site

Abstract

TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) by JIAN DING ,JULIEN DUBÉDAT ,ALEXANDER DUNLAP , and HUGO FALCONET ABSTRACT We study Liouville ﬁrst passage percolation metrics associated to a Gaussian free ﬁeld h molliﬁed by the two- dimensional heat kernel p in the bulk, and related star-scale invariant metrics. For γ ∈ (0, 2) and ξ = ,where d is t γ the Liouville quantum gravity dimension deﬁned in Ding and Gwynne (Commun. Math. Phys. 374:1877–1934, 2020), −1 ξ p ∗h we show that renormalized metrics (λ e ds) are tight with respect to the uniform topology. We also show that t∈(0,1) subsequential limits are bi-Hölder with respect to the Euclidean metric, obtain tail estimates for side-to-side distances, and derive error bounds for the normalizing constants λ . CONTENTS 1. Introduction and main statement ........................................... 353 1.1. Strategy of the proof and comparison with previous works ........................... 357 2. Description and comparison of approximations ................................... 359 2.1. Basic properties of φ and ψ ........................................... 360 δ δ 2.2. Comparison between φ and ψ ......................................... 365 δ δ 2.3. Length observables ................................................ 366 2.4. Outline of the proof and roles of φ and ψ ................................... 367 δ δ 3. Russo–Seymour–Welsh estimates ........................................... 367 3.1. Approximate conformal invariance ....................................... 367 3.2. Russo–Seymour–Welsh estimates ......................................... 371 4. Tail estimates with respect to ﬁxed quantiles ..................................... 376 5. Concentration ...................................................... 380 5.1. Concentration of the log of the left–right crossing length ............................ 380 5.2. Weak multiplicativity of the characteristic length and error bounds ...................... 388 5.3. Tightness of the log of the diameter ....................................... 391 5.4. Tightness of the metrics .............................................. 393 Acknowledgements ..................................................... 396 Publisher’s Note ...................................................... 396 Appendix A . ........................................................ 396 A.1. Comparison with the GFF molliﬁed by the heat kernel ............................. 396 A.2. Approximations for δ ∈ (0, 1) .......................................... 400 References ......................................................... 401 1. Introduction and main statement We consider the problem of rigorously constructing a metric for Liouville quantum γ h gravity (LQG), a random geometry formally given by reweighting Euclidean space by e , where h is a Gaussian free ﬁeld. LQG was originally introduced in the physics literature by Polyakov in 1981 [40]. In its mathematical form, the LQG measure is a special case of Gaussian multiplicative chaos, introduced in [29]. In the last two decades, there has been J. Ding was partially supported by NSF grant DMS-1757479 and an Alfred Sloan fellowship. J. Dubédat was Partially supported by NSF grant DMS-1512853. A. Dunlap was partially supported by an NSF Graduate Research Fellowship. © IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2020 https://doi.org/10.1007/s10240-020-00121-1 354 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET an explosion of interest in the probability community towards rigorously constructing the relevant objects. In particular, the LQG measure was constructed rigorously in the regime γ ≤ 2, via a renormalization procedure, in [21]. Other relevant work in this area includes [2, 3, 41–44, 48]. Much remains open regarding the construction of the LQG metric. When γ = 8/3, LQG is intimately connected with the Brownian map [31, 32, 35]and ametric for LQG has been constructed in [36–38]. Substantial work has also been devoted to understanding the distance exponents for natural discrete LQG metrics; see [1, 11, 12, 15, 27, 28]. In [14, 16] some non-universality results were established for ﬁrst-passage γφ percolation distance exponents for metrics of the form e ds,where φ is discretization of a log-correlated Gaussian ﬁeld. This indicates that precisely understanding such expo- nents must involve rather ﬁne information about the structure of the particular ﬁeld in question. The present study concerns the tightness of Liouville ﬁrst-passage percolation (LFPP) metrics, which are natural smoothed LQG metrics. This proves the existence of subsequential limiting metrics. Given this, it remains to show that such limiting metrics are unique in law for each γ ∈ (0, 2) in order to complete the construction of the LQG metric in this regime. After this paper was posted, the latter task was carried out in the series of works [19, 23–26], thus completing the construction. The present study follows three main tightness results for discretized or smoothed LQG metrics. In [9], tightness of LFPP metrics (on a discrete lattice) was proved in the small noise regime for which γ is very small. In [18], tightness was shown for metrics arising in the same way from -scale invariant ﬁelds, still in the small noise regime. In [10], tightness was shown for all γ< 2 for the Liouville graph distance, which is a graph metric equal to the least number of Euclidean balls of a given LQG measure necessary to cover a path between a pair of points. We consider a smoothed Gaussian ﬁeld (1.1) φ (x) := π p (x − y)W(dy, dt) 2 2 δ R |x−y| 2 1 − 2t for x ∈ R and δ ∈ (0, 1),where p (x − y) := e and W is a space–time white 2π t noise. This approximation is natural since it can be uniformly compared on a compact domain with a Gaussian free ﬁeld h molliﬁed by the heat kernel deﬁned on a slightly larger domain, viz. φ and p ∗ h (where ∗ denotes the convolution operator) are comparable. t t/2 Furthermore, this approximation provides some nice invariance and scaling properties on the full plane. For γ ∈ (0, 2),wewilluse thenotation (1.2) ξ := γ/d where d is the “Liouville quantum gravity dimension” deﬁned in [12]. It is known (see Theorem 1.2 and Proposition 1.7 in [12]) that the function γ → γ/d is strictly increasing γ TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 355 and continuous on (0, 2). Therefore, in this article we will be interested in the range − − ξ ∈ (0,(2/d ) ),where (2/d ) = lim γ/d . 2 2 γ ↑2 γ ξφ We consider the length metric e ds (equivalently, the metric whose Riemannian 2ξφ 2 2 metric tensor is given by e ds ), restricted to the unit square [0, 1] .Werecallthat a length metric is a metric such that the distance between two points is given by the inﬁmum over the arc lengths of paths connecting the two points. We denote by λ the 2 ξφ median of the left–right distance of [0, 1] for the metric e ds. Our main theorem is the following. −1 ξφ Theorem 1.—1. If γ ∈ (0, 2), then λ e ds is tight with respect to the δ∈(0,1) 2 2 + uniform topology on the space of continuous functions [0, 1] ×[0, 1] → R .Furthermore, any subsequential limit is almost surely bi-Hölder with respect to the Euclidean metric on [0, 1] . 2. Let K =[0, 1] . If h is a Gaussian free ﬁeld with zero boundary conditions on a bounded open domain D containing K (extended to zero outside of D), then the internal metrics ξ p ∗h −1 √ 2 (λ e ds) on K are tight with respect to the uniform topology of continuous func- δ∈(0,1) tions K × K → R . Furthermore, the normalizing constants (λ ) satisfy δ δ∈(0,1) O | log δ| 1−ξ Q (1.3) λ = δ e where Q = + . γ 2 A year after our article was posted, the subsequent work [13]provedasimilar result to ours when ξ ≥ (2/d ) . However, in that case the tightness does not hold in the uniform topology and the Beer topology on lower semicontinuous functions was used. In order to establish the tightness of the family of renormalized metrics −1 ξφ (d ) := (λ e ds) , we prove a number of uniform estimates for that fam- φ δ∈(0,1) δ∈(0,1) δ δ ily (which also hold when the approximation is the GFF molliﬁed by the heat kernel). Such estimates that are closed under weak convergence also apply to subsequential lim- its. Let us summarize these properties. Let D denote the family of laws of d , δ ∈ (0, 1) 2 2 (i.e. seen as random continuous functions on ([0, 1] ) ), and D denotes its closure under weak convergence (i.e., D also includes the laws of all subsequential limits). 1. Under any P ∈ D, d is P-a.s. a length metric. This is clear for the renormalized metrics d by deﬁnition, and the property of being a length metric extends to limits. (See [6, Exercise 2.4.19].) 2. If d is a metric on R and R is a rectangle, we denote by d(R) the left–right length of R for d . We have the following tail estimates. There exists c, C > 0 356 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET such that for s > 2, uniformly in P ∈ D we have 2 2 −Cs −s −cs (1.4) ce ≤ P d(R) ≤ e ≤ Ce , −c −Cs s log s (1.5) ce ≤ P (d(R) ≥ e ) ≤ Ce . The upper bounds are proved in Section 4, while the lower bounds are conse- quences of the Cameron–Martin theorem, considering shifts of the ﬁeld at the coarsest scale as in [18, Section 5.4]. 3. If d is a metric on R and R is a rectangle, we denote by Diam(R, d) the diam- eter of R for d . We have the following uniform ﬁrst moment bound: (1.6)sup E (Diam(R, d)) < ∞. P∈D This is shown in the course of the proof of Proposition 27 below. 4. Under any P ∈ D, d is P-a.s. bi-Hölder with respect to the Euclidean metric and we have the following bounds for exponents: for α< ξ(Q − 2), β> ξ(Q + 2), and R a rectangle, the families |x − x | d(x, x ) (1.7) sup and sup d(x, x ) |x − x | x,x ∈R x,x ∈R L(d)∈D L(d)∈D are tight. Here L(d) means the law of d . These properties are shown in Propo- sition 28 below. Let us also mention that subsequential limits are consistent with the Weyl scal- ing: for a function f in the Cameron–Martin space of the Gaussian free ﬁeld h,for any coupling (h, d) associated to a subsequential limit of the sequence of laws of ξ p ∗h −1 ξ f √ 2 ((h,λ e ds)) , the couplings (h, d) and (h + f , e · d) are mutually absolutely con- δ>0 tinuous with respect to each other and the associated Radon–Nikodým derivative is the one of the ﬁrst marginal. This can be proved using similar arguments to those of [18, Section 7]. An analogue of this property for the Liouville measure together with the con- servation of the Liouville volume average is enough to characterize the Liouville measure, as seen by Shamov in [48]. It may be interesting to draw a parallel between our work and those in random planar maps, since both aim to obtain the scaling limits of random metrics and the lim- iting objects are related for γ = 8/3. We start with a brief overview on the conver- 1/4 gence of random planar maps. Chassaing and Schaeffer [7] identiﬁed n as the proper scaling and compute certain limiting functionals for random quadrangulations. Marck- ert and Mokkadem [34] established limit theorems (in a sense weaker than Gromov– Hausdorff) and introduced the Brownian map. Le Gall [30] showed tightness for rescaled 2p-angulations in the Gromov–Hausdorff topology and shows that the limiting topology TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 357 is the same as the Brownian map. Le Gall and Paulin [33] showed that the limiting topol- ogy is that of the 2-sphere and Le Gall studied properties of geodesics in [31]. Finally, Le Gall [32] (resp. Miermont [35]) proved the uniqueness of subsequential limits of uniform triangulations and 2p-angulations (resp. quadrangulations). In both cases, the proof relied on a careful study of geodesics and in particular on conﬂuence properties, together with a rough bound quantifying an approximate equivalence of the two metrics to match. In our framework, an important result was obtained in [12], where the authors identiﬁed the exponent of LFPP distances to be 1 − ξ Q + o(1). By contrast, in the ran- 1/4 dom planar map setting, the normalization is exactly n and Chassaing and Schaeffer [7] obtained the convergence in law of some observables. In our case, the tightness of any observable renormalized by its median is far from obvious. This is a common feature of some concentration problems for extrema of random ﬁelds. As an analogy, one can consider the problem of the tightness of the maxima of branching random walks (BRW), where, also by subadditivity, the expected value of the maxima of BRW on a d -ary tree ∗ ∗ at level n is of order (x + o(1))n for some x (which depends on the rate function of the distribution of the increments). A powerful and well-understood method in proving tight- ness for BRW is by an explicit truncated second moment estimate which computes the expected maxima up to additive O(1) constant (see [4] for maxima of BRW and [5]for maxima of discrete GFF). In contrast, in our setup, explicit computation on the distance seems really difﬁcult; in fact it remains a major challenge to compute the value of the dis- tance exponent, let alone computing the distance up to constant. In order to circumvent this difﬁculty, we had to build our proof by exploring delicate intrinsic structure of the distance. We point out here that it was shown in [1]thatfor γ ∈ (0, 2),1 − ξ Q ≥ 0(and it is believed to be > 0), therefore the normalization λ should be thought as small. Furthermore, in our setting where the metrics are on a compact subset of C,we can directly use the uniform topology instead of working with the Gromov–Hausdorff topology (note that the former is stronger than the latter). In this paper, we show tight- ness for the full subcritical range γ ∈ (0, 2) of renormalized side-to-side crossing lengths, point-to-point distance and metrics. Limiting metrics are bi-Hölder with respect to the Euclidean metric. 1.1. Strategy of the proof and comparison with previous works. — In contrast with previous works on the LQG measure, the variational problem deﬁning the LQG metric means that most direct computations are impossible, and in particular most of techniques used in the theory of Gaussian multiplicative chaos and LQG measure are unavailable. This necessitates the more intricate multiscale geometric arguments that we employ. Our tightness proof relies on two key ingredients, a Russo–Seymour–Welsh argu- ment and multiscale analysis. In both parts we extend and reﬁne many arguments used in the previous works [9, 10, 18] on the tightness of various types of LQG metrics. Russo–Seymour–Welsh. — The RSW argument relates, to within a constant factor, quan- tiles of the left–right LFPP crossing distances of a “portrait” rectangle and of a “land- 358 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET scape” rectangle. (By a crossing distance we simply mean the distance between two op- posite sides of a rectangle.) In [9, 10], these crossings are referred to as “easy” and “hard” respectively. The utility of such a result is that crossings of larger rectangles necessarily induce easy crossings of subrectangles, while hard crossings of smaller rectangles can be glued together to create crossings of larger rectangles. Thus, multiscale analysis argu- ments can establish lower bounds in terms of easy crossings and upper bounds in terms of hard crossings. RSW arguments then allow these bounds to be compared. RSW arguments originated in the works [45–47] for Bernoulli percolation, and have since been adapted to many percolation settings. The work [9] introduced an RSW result for LFPP in the small noise regime based on an RSW result for Voronoi percolation devised by Tassion [49]. Tassion’s result is beautiful but intricate, and becomes quite complex when it is adapted to take into account the weights of crossing in the ﬁrst-passage percolation setting, as was done in [9]. The RSW approach of this paper is based on the much simpler approach intro- duced in [18], which relies on an approximate conformal invariance of the ﬁeld. (We recall that the Gaussian free ﬁeld is exactly conformally invariant in dimension 2, and that the LQG measure enjoys an exact conformal covariance.) Roughly speaking, the conformal invariance argument relies on writing down a conformal map between the portrait and landscape rectangles, and analyzing the effect of such a map on crossings of the rectangle. We note that the approximate conformal invariance used in this paper relies in an important way on the exact independence of different “scales” of the ﬁeld, which is manifest in the independence of the white noise at different times in the expres- sion (1.1). Thus, the argument we use here is not immediately applicable to molliﬁcations of the Gaussian free ﬁeld by general molliﬁers (for example, the common “circle-average approximation” of the GFF). The RSW argument of [18] was also adapted in [10]tothe Liouville graph distance case. Tail estimates. — Once the RSW result is established, we derive tail estimates with respect to ﬁxed quantiles. The lower tail estimate is unconditional, while the upper tail estimate depends on a quantity measuring the concentration at the current scale, which will later be uniformly bounded by an inductive argument. Multiscale analysis. — With RSW and tail estimates in hand, we turn to the multiscale analysis part of the paper. This argument turns on the Condition (T) formulated in (5.2) below, which, informally, states that the arclength of the crossing is not concentrated on a small number of subarcs of small Euclidean diameter. The argument of [10] requires similar input, which is a key role of the subcriticality γ< 2. While [10] relies directly on certain scaling symmetries of the Liouville graph distance to use subcriticality, the present work relies on the characterization of the Hausdorff dimension d obtained in [12], along with some weak multiplicativity arguments and concentration obtained from percolation arguments. TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 359 Condition (T). — Our formulation of Condition (T), which has not appeared in previous works, precisely captures the property of the metric needed to obtain the tightness of the left–right crossing distances, the existence of the exponent, and the tail estimates (via a uniform bound on the ). Condition (T) makes sense for LFPP with any underlying ﬁeld and any parame- ter ξ . In particular, this condition or a variant thereof could possibly hold for LFPP for some ξ> 2/d . Therefore, a byproduct of the present work is a simple criterion (that im- plies, as noted above, tightness of the crossing distances, existence of exponents, and tail estimates) that may be applicable more generally. The utility of Condition (T) is that it allows us to use an Efron–Stein argument to obtain a contraction in an inductive bound on the crossing distance logarithm variance. Informally, since the crossing distance feels the effect of many different subboxes, the subbox crossing distances are effectively being averaged to form the overall crossing dis- tance. This yields a contraction in variance. (Of course, the coarse scales also contribute to the variance, and hence the variance of the crossing distance does not decrease as the discretization scale decreases but rather stays bounded.) The way we verify Condition (T) is quite rough: we bound the ﬁeld uniformly over a coarse grained geodesic by the supremum of the ﬁeld over the unit square. It turns out that this bound together with the identiﬁcation of the exponent 1 − ξQis enough to establish the condition. Tightness of the metrics. — Once the tightness of the left–right crossing distance is estab- lished, we turn to the tightness of the diameter and of the metric itself. This is done by a chaining argument, and requires again ξ< 2/d . The diameter is not expected to be tight when ξ> 2/d , since there are points that become inﬁnitely distant from the bulk of the space as the discretization scale goes to 0. 2. Description and comparison of approximations We recall that a white noise W on R is a random Schwartz distribution such that for every smooth and compactly supported test function f , W, f is a centered Gaussian variable with variance f 2 d (see e.g. [8]). The main approximation of the Gaussian L (R ) free ﬁeld that we consider in this paper is deﬁned for δ ∈ (0, 1) by (2.1) φ (x) := π p (x − y)W(dy, dt) 2 2 δ R |x−y| 1 − 2 2t where p (x − y) := e and W is a space–time white noise on [0, 1]× R . This 2π t approximation is different than the one considered in [18] which is x − y −3/2 φ (x) := k t W(dy, dt) 2 t δ R 360 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET for a smooth nonnegative bump function k, radially symmetric and with compact sup- port. Up to a change of variable in t, the difference is essentially replacing p by k.Both ﬁelds are normalized in such a way that E(φ (x)φ (y)) =− log |x − y|+ g(x, y) with g 0 0 continuous (see e.g. Section 2 in [18]): this is the reason for the factor π in (2.1). Let us mention that -scale invariant Gaussian ﬁelds with compactly-supported bump function k 1. are invariant under Euclidean isometries, 2. have ﬁnite-range correlation at each scale, 3. and have convenient scaling properties. The Gaussian ﬁeld φ introduced above satisﬁes 1 and 3 but not 2. Because of the lack of ﬁnite-range correlation, we will also use a ﬁeld ψ (deﬁned in the next section) which satisﬁes 1 and 2 such that sup φ − ψ has Gaussian tails, where we use 0,n 0,n ∞ 2 n≥0 L ([0,1] ) −n the notation φ for φ with δ = 2 . 0,n δ 2.1. Basic properties of φ and ψ . δ δ Scaling property of φ .— We use the scale decomposition −2n φ := φ where φ (x) = π p (x − y)W(dy, dt) n n −2(n+1) 2 2 R n≥0 If we denote by C the covariance kernel of φ ,so C (x, x ) = E(φ (x)φ (x )),thenwe n n n n n have −2n 1 |x−x | − n n 2t C (x, x ) = e dt = C (2 x, 2 x ). n 0 −2(n+1) 2t Therefore, the law of (φ (x)) 2 is the same as (φ (2 x)) 2 . Because of the above, n x∈[0,1] 0 x∈[0,1] 2t 2 −1 we choose δ and not δ in (2.1) so that the pointwise variance φ is log δ . Similarly, for 0 < a < b and x ∈ R , set (2.2) φ (x) := π p (x − y)W(dy, dt) a,b 2 2 a R (d) and note that we have the scaling identity φ (r·) = φ (·). Indeed, we have a,b a/r,b/r t t E(φ (rx)φ (rx )) = π p (rx − y)p (y − rx )dydt a,b a,b 2 2 2 2 a R 2 2 b b 2 2 1 r |x−x | 2t = π p (r(x − x ))dt = e dt, 2 2 2t a a TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 361 and by the change of variable t = r u, this gives 2 2 b (b/r) 2 2 2 1 r |x−x | 1 |x−x | − − 2t 2t e dt = e dt = E(φ (x)φ (x )). a/r,b/r a/r,b/r 2 2t 2 2t a (a/r) −n −k We will use the notation φ when a = 2 and b = 2 for 0 ≤ k ≤ n. k,n Maximum and oscillation of φ .— We have the same estimates for the supremum of the ﬁeld φ as those for the -scale invariant case considered in [18] (it is essentially a union 0,n bound combined with a scaling argument). The following proposition corresponds to Lemma 10.1 and Lemma 10.2 in [18]. Proposition 2 (Maximum bounds).— We have the following tail estimates for the supremum of φ over the unit square: for a > 0,n ≥ 0, 0,n √ a − n log 4 (2.3) P max |φ |≥ a(n + C n) ≤ C4 e 0,n [0,1] as well as the following moment bound: if γ< 2, then γ max |φ | γ n+O( n) 2 0,n [0,1] (2.4) E(e ) ≤ 4 We will also need some control on the oscillation of the ﬁeld φ . We introduce 0,n ∞ d d the following notation for the L -norm on a subset of R . If A is a subset of R and f : A → R , we set (2.5) f := sup |f (x)| x∈A We introduce the following notation to describe the oscillation of a smooth ﬁeld φ:if A ⊂ R we set (2.6)osc (φ) := diam(A) ∇φ , so that if A is convex then sup |φ(x) − φ(y)|≤ osc (φ) and x,y∈A −n max osc (φ ) ≤ C2 ∇φ , P 0,n 0,n [0,1] P∈P ,P⊂[0,1] where P denotes the set of dyadic blocks at scale n,viz. −n (2.7) P := {2 ([i, i + 1]×[j, j + 1]) : i, j ∈ Z}. In order to simplify the notation P ∈ P ,P ⊂[0, 1] later on, we also set 1 2 (2.8) P := {P ∈ P : P ⊂[0, 1] }. n 362 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Proposition 3 (Oscillation bounds).— We have the following tail estimates for the oscillation of φ : there exists C > 0, σ > 0, so that, for all x,ε > 0,n ≥ 0, 0,n −n n 2σ (2.9) P 2 ∇φ ≥ x ≤ C4 e 0,n 2 [0,1] as well as the following moment bound: for a > 0, there exists c > 0 so that for n ≥ 0, ε −n +ε 2ε an 2 ∇φ 2 0,n c n +O(n ) 2 a [0,1] (2.10) E e ≤ e Proof. — Inequality (2.9) was obtained between Equation (10.3) and Equation ε −n (10.4) in [18]. Now, we prove (2.10). Set a := an ,O = 2 ∇φ ,and take n n 0,n [0,1] 2 α x = a σ + ασ n with α> 0so that = log 4. We have, using (2.9), n n ∞ ∞ ∞ a O a O s s n 2 s n n n n 2a σ P e ≥ x dx = P e ≥ e e ds ≤ C4 e e ds a x n n e a x a x n n n n By a change of variable (s ↔ a σ s + (a σ) ), we get n n ∞ 2 ∞ ∞ s 2 2 − 1 2 2 s 1 2 2 s 2 s a σ − a σ − 2a σ 2 n 2 2 n 2 e e ds = a σ e e ds = a σ e e ds n n xn a x −a σ α n n n n 2 2 2 −bx −1 −ba since x = a σ + ασ n. Using that e dx ≤ (2ab) e ,weget n n 2ε a O O(n ) a O a x n n n n n n P e ≥ x dx ≤ e . The result follows from writing E(e ) ≤ e + a x n n a O n n P(e ≥ x)dx. a x n n Deﬁnition of ψ .— We ﬁx a smooth, nonnegative, radially symmetric bump function such that 0 ≤ ≤ 1and is equal to one on B(0, 1) and to zero outside B(0, 2).We also ﬁx small constants r > 0and ε > 0. We will specify these constants later on. In 0 0 particular, ε appears in the main proof in (5.11) and its ﬁnal effect is in (5.16). All other constants C, c will implicitly depend on r and ε . Then, we introduce for each δ ∈[0, 1], 0 0 the ﬁeld 1 1 Tr (2.11) ψ (x) := (x−y)p (x−y)W(dy, dt) = p (x − y)W(dy, dt) δ σ 2 2 2 2 δ R δ R ε Tr where σ = r t| log t| , (·) := (·/σ ) and p := p t . t 0 σ t σ t t Thanks to the truncation, the ﬁelds (ψ ) have ﬁnite correlation length δ δ∈[0,1] 8r sup t| log t| . t∈[0,1] TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 363 Decomposition in scales and blocks of ψ .— We have the scale decomposition Tr (2.12) ψ(x) := p (x − y)W(dy, dt) 0 R −2k+2 Tr = p t (x − y)W(dy, dt) = ψ (x) k,P −2k 2 P k=1 k≥1 P∈P P∈P k k −2k+2 Tr where ψ is deﬁned for P ∈ P by ψ (x) := p (x − y)W(dy, dt) and thus has k,P k k,P −2k 2 P ε −k correlation length less than Ck 2 . In particular, a ﬁxed block ﬁeld is only correlated 2ε with fewer than Ck other block ﬁelds at the same scale. In fact, when we apply the Efron–Stein inequality (see (5.9)) we will use the following decomposition: (2.13) ψ = ψ + ψ (x) 0,n 0,K K,n,P P∈P −2K+2 Tr where ψ (x) := p (x − y)W(dy, dt). K,n,P −2n 2 P We note that there is a formal conﬂict in notation between (2.2)and (2.13), butitwill always be clear from context whether the second subscript is a number or an element of P (a set), so confusion should not arise. Variance bounds for φ and ψ .— Later on we will need the following lemma. δ δ Lemma 4. — There exists C > 0 so that for δ ∈[0, 1] and x, x ∈ R , we have |x − x | (2.14)Var φ (x) − φ (x ) + Var ψ (x) − ψ (x ) ≤ C . δ δ δ δ −z Proof. — We start by estimating the ﬁrst term. Using the inequality 1 − e ≤ z ≤ √ √ −z z for z ∈[0, 1] and 1 − e ≤ 1 ≤ z for z ≥ 1we get t t t t Var φ (x) − φ (x ) = C p ∗ p (0) − p ∗ p (x − x ) dt δ δ 2 2 2 2 = C p (0) − p (x − x ) dt t t 1 |x−x | 2t = C (1 − e )dt 2 t dt |x − x | ≤ C|x − x | = C . 3/2 2 t δ δ 364 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Similarly, for the second term, we have Tr Tr Tr Tr Var ψ (x) − ψ (x ) = C p t ∗ p t (0) − p t ∗ p t (x − x ) dt. δ δ 2 2 2 2 Tr Tr Set p ∗ p =: p (x)q (x). Using the identity p (y)p (x − y) = p (x)p (y − x/2) we get t t t t t/2 t/2 t t/4 2 2 p (y)p (x − y) t/2 t/2 q (x) = (y) (x − y)dy t σ σ t t 2 p (x) = p (y − x/2) (y) (x − y)dy. t/4 σ σ t t We rewrite the variance in terms of q : replacing x − x by z we look at Var ψ (x) − ψ (x ) δ δ = C (p (0)q (0) − p (z)q (z))dt t t t t 1 1 = C p (0)(q (0) − q (z))dt + C q (z)(p (0) − p (z))dt. t t t t t t 2 2 δ δ We deal with these two terms separately. For the second one, since 0 ≤ ≤ 1, we have 0 ≤ q ≤ 1. Therefore, following what we did for φ above we directly have t δ |z| −1 0 ≤ q (z)(p (0) − p (z))dt ≤ C . For the ﬁrst term, since p (0) = Ct ,itisenough 2 t t t t to get the bound t|q (0) − q (z)|≤ C|z| to complete the proof of the lemma. Changing t t variables, we have √ √ −2|y| q (z) = C e ( ty + z/2) ( ty − z/2)dy. t σ σ t t Therefore, using that 0 ≤ ≤ 1, −2|y| |q (z) − q (0)|≤ C e | ( ty + z/2) − ( ty)|dy t t σ σ t t −2|y| + C e | ( ty − z/2) − ( ty)|dy σ σ t t 2 |z| 2 −2|y| −2|y| ≤ C|z| e ∇ dy ≤ C ∇ 2 e dy. σ 2 R 2 σ 2 R t R ε t Since σ = r t| log t| , we see that sup < ∞, and the result follows. t 0 t∈[0,1] t TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 365 2.2. Comparison between φ and ψ .— The following proposition justiﬁes the intro- δ δ duction of the ﬁeld ψ . Proposition 5. — There exist C > 0 and c > 0 such that for all x > 0, we have −cx (2.15) P sup φ − ψ ≥ x ≤ Ce . 0,n 0,n [0,1] n≥0 Proof. — For k ≥ 1, we introduce the quantity D (x) := φ (x) − ψ (x).The k k−1,k k−1,k proof follows from an adaptation of Lemma 2.7 in [17] as soon as we have the estimates 2ε −ck (2.16)VarD (x) ≤ Ce and (2.17)Var φ (x) − φ (y) + Var ψ (x) − ψ (y) ≤ 2 |x − y|. k k k k (The estimate (2.16) is weaker than that used in [17, Lemma 2.7] but still much stronger than required for the proof given there.) Note that (2.17) follows from Lemma 4 and for (2.16) we proceed as follows: ﬁrst note that −2k+2 2 2 E φ (x) − ψ (x) = p (y) (1 − (y)) dydt. k−1,k k−1,k σ −2k 2 2 R −1 −σ /t For every y,wehave p (y)(1 − (y)) ≤ (2π t) e since 0 ≤ ≤ 1and (y) = 1 t/2 σ σ σ t t t for |y|≤ σ . Therefore, −2k+2 2 2ε −ck E φ (x) − ψ (x) ≤ p (y)dydt ≤ Ce . k−1,k k−1,k −2k 2π t 2 2 R Let us point out that in fact E( φ − ψ )< ∞ holds but we n,n+1 n,n+1 n≥0 [0,1] won’t use it. Since we will be working with two different approximations of the Gaus- sian free ﬁeld, we introduce here some notation, referring to one ﬁeld or the other. We will denote by R := [0, a]×[0, b] the rectangle of size (a, b).Wedeﬁne a,b (2.18)X := sup φ − ψ a,b 0,n 0,n a,b n≥0 and X := X for the supremum norm of the difference between the two ﬁelds on vari- a a,a ous rectangles. 366 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET (n) (n) 2.3. Length observables. — The symbol L (φ) (and similarly L (ψ )) will refer to a,b a,b ξφ 0,n the left–right distance of the rectangle R for the length functional e ds: a,b (n) ξφ 0,n (2.19)L (φ) := inf e ds, a,b where ds refers to the Euclidean length measure and the inﬁmum is taken over all smooth curves π connecting the left and right sides of R . We will sometimes consider a geodesic a,b associated to this variational problem. Such a path exists by the Hopf–Rinow theorem and a compactness argument. We introduce some notation for the quantiles associated to this observable: (n) (n) (n) (n) (φ, p) (similarly (ψ, p))issuchthat P L (φ) ≤ (φ) = p. For high quantiles, a,b a,b a,b a,b (n) (n) (n) we introduce (φ, p) := (φ, 1 − p).Notethat (φ, p) is increasing in p whereas a,b a,b a,b (n) (φ, p) is decreasing in p. Note that both are well-deﬁned, i.e., there are no Dirac deltas a,b (n) in the law of L . This follows from an application of the Cameron–Martin formula. We a,b will also need the notation (φ, p) (2.20) (φ, p) := max k≤n (φ, p) (k) (k) ¯ ¯ where (φ, p) := (φ, p) and (φ, p) := (φ, p). k k 1,1 1,1 The following inequalities are straightforward: −ξ X (n) (n) ξ X (n) a,b a,b (2.21) e L (ψ ) ≤ L (φ) ≤ e L (ψ ) a,b a,b a,b Therefore, using Proposition 5 (and a union bound, if necessary), we obtain that for some C > 0 (depending only on a and b), for any ε> 0we have √ √ (n) (n) (n) −ξ C | log ε/C| ξ C | log ε/C| ¯ ¯ ¯ e (ψ, p + ε) ≤ (φ, p) ≤ e (ψ, p − ε) a,b a,b a,b √ √ (n) (n) (n) −ξ C | log ε/C| ξ C | log ε/C| e (ψ, p − ε) ≤ (φ, p) ≤ e (ψ, p + ε) a,b a,b a,b In particular, there exists C > 0 such that, uniformly in n, −1 ¯ ¯ (ψ, p/2) ≥ C (φ, p), (ψ, p/2) ≤ C (φ, p) and n p n n p n (2.22) (ψ, p/2) ≤ C (φ, p). n p n Now, we discuss how the scaling property of the ﬁeld φ translates at the level of lengths. We will use the following equality in law: for a, b > 0and 0 ≤ m ≤ n, (d) (m,n) (n−m) −m (2.23)L (φ) = 2 L (φ). m m a,b a·2 ,b·2 TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 367 (n) Finally, for a rectangle P with two marked opposite sides, we deﬁne L (P,φ) to ξφ 0,n be the crossing distance between the two marked sides under the ﬁeld e . The marked sides will be clear from context: if we call P a “long rectangle,” then we mean that the (n) marked sides are the two shorter sides, so that L (P,φ) is the distance across P “the long way.” 2.4. Outline of the proof and roles of φ and ψ .— The key idea of the proof is to δ δ obtain a self-bounding estimate associated to a measure of concentration of some ob- servables, say rectangle crossing lengths. This is naturally expected because of the tree structure of our model. We introduce a general condition, which we call Condition (T), (see (5.2)) which ensures a contraction in the self-bounding estimate (5.19), which relates a measure of concentration at scale n, the variance, with the measure of concentration that we inductively bound, (see (2.20)), which is at a smaller scale. n−K We then prove that this condition, which depends only on ξ and on the ﬁeld con- sidered, is satisﬁed when ξ ∈ (0,(2/d ) ). This proof uses a result taken from [12] about the existence of an exponent for circle average Liouville ﬁrst passage percolation and this is the reason we don’t consider the simpler -scale invariant ﬁeld with compactly- supported kernel but the ﬁeld φ , which can be compared to the circle average process by a result obtained in [11]. The roles of φ and ψ in the proof are the following. δ δ 1. Prove Russo–Seymour–Welsh estimates for φ. 2. Prove tail estimates w.r.t. low and high quantiles for both φ and ψ : a) Lower tails: Use directly the RSW estimates together with a Fernique-type argument for the ﬁeld ψ with local independence properties. b) Upper tails: use a percolation/scaling argument, percolation using ψ and scaling using φ. 3. Concentration of the log of the left–right distance: use Efron–Stein for the ﬁeld ψ (because of the local independence properties at each scale). This gives the same result for φ. 4. To conclude for the concentration of diameter and metric, this is essentially a chaining/scaling argument using only the ﬁeld φ. 3. Russo–Seymour–Welsh estimates 3.1. Approximate conformal invariance. — In order to establish our RSW result, we ﬁrst show an approximate conformal invariance property of the ﬁeld. The arguments in this section are similar to those of [18, Section 3.1]. The difference is that the Gaussian kernel has inﬁnite support. |x−y| t 2t Recall that φ (x) = p (x − y)W(dy, dt) where p (x − y) = e .Consider δ 2 2 t δ R 2π t a conformal map F between two bounded, convex, simply-connected open sets U and 368 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Vsuch that |F | ≥ 1onU, F < ∞ and F < ∞. (We point out here that the U U assumption |F |≥ 1 will be obtained later on by starting from a very small domain; this is exactly the content of Lemma 11.) We consider another ﬁeld φ (x) = p (x − δ 2 2 δ R ˜ ˜ y)W(dy, dt) where W is a white noise that we will couple with W in order to compare φ and φ ◦ F. The coupling goes as follows: for y ∈ U, t ∈ (0, ∞),let y = F(y) ∈ Vand δ δ 2 2 2 t = t|F (y)| and set W(dy , dt ) =|F (y)| W(dy, dt).Thatis, forevery L function ω on V × (0, ∞), ω(y , t )W(dy , dt ) = ω(F(y), t|F (y)| ) F (y) W(dy, dt) and both sides have variance ω . The rest of the white noises are chosen to be inde- c c pendent, i.e., W ,W and W are jointly independent. |U ×(0,∞) |U×(0,∞) |V ×(0,∞) Lemma 6. — Under this coupling, we can compare the two ﬁelds φ (F(x)) and φ (x) on a δ δ compact, convex subset K of U as follows, (δ) (δ) (3.1) φ (F(x)) − φ (x) = φ (x) + φ (x), δ δ L H (δ) where φ (L for low frequency noise) is a smooth Gaussian ﬁeld whose L -norm on K has uniform (δ) Gaussian tails, and φ (H for high frequency noise) is a smooth Gaussian ﬁeld with uniformly bounded (δ) (δ) pointwise variance (in δ and x ∈ K). Furthermore, φ is independent of (φ ,φ ). H L This aforementioned independence property will be crucial for our argument. Proof. — Step 1: Decomposition. For ﬁxed F and small δ, we decompose φ (x) − (δ) (δ) (δ) φ (F(x)) = φ (x) + φ (x) + φ (x),where 1 2 3 −2 F (y) | | (δ) φ (x) = p (x−y) − p 2 F(x)−F(y) F (y) W(dy, dt) F (y) 2 | | U δ −2 F (y) | | F(x) − F(y) t t = p (x − y) − p W(dy, dt) 2 2 2 F (y) U δ 1 1 (δ) t t φ (x) = p (x − y)W(dy, dt) − p F(x) − y W(dy, dt) 2 2 c 2 c 2 U δ V δ + p t (x − y)W(dy, dt) −2 U F (y) | | F(x) − F(y) (δ) φ (x) =− p W(dy, dt) −2 2 F (y) U δ F (y) | | (δ) (δ) (δ) Remark also that φ is independent of φ , φ ,and φ . 3 1 2 TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 369 (δ) Step 2: Conclusion, assuming uniform estimates. We will estimate φ , i = 1, 2, 3, over K. In what follows, we take x, x ∈ K. We assume ﬁrst the following uniform esti- mates: (δ) (δ) E((φ (x) − φ (x )) ) ≤ C x − x , 1 1 (δ) (δ) (δ) E((φ (x) − φ (x )) ) ≤ C x − x , E φ (x) ≤ C. 2 2 3 An application of Kolmogorov’s continuity criterion and Fernique’s theorem give uni- (δ) (δ) (δ) (δ) (δ) (δ) (δ) form Gaussian tails for φ and φ . We then set φ := φ and φ := φ + φ . 1 2 H 3 L 1 2 Step 3: Uniform estimates. (δ) (δ) First term. We prove that E((φ (x) − φ (x )) ) ≤ C |x − x | by controlling 1 1 F(x) − F(u) t t p (x − y) − p 2 2 F (y) 0 U F(x ) − F(u) t t − p x − y + p dydt 2 2 F (y) |x| − 2 By introducing p(x) = e and by a change of variable t ↔ 2t , it is equivalent (up to a multiplicative constant) to bound from above the quantity dt x − y F(x) − F(y) (3.2) p − p t t tF (y) 0 U x − y F(x ) − F(y) − p + p dy. t tF (y) We will estimate this term by considering the case where t ≤ |x − x | and the case where t ≥ |x − x |. 2 2 1 2 Step 3.(A): Case t ≥ |x − x |. Using the identity |x − y| +|x − y| = |x − x | + x+x 2 −z 2|y − | and the inequality 1 − e ≤ z,weget |x−x | x − y x − y 4t (3.3) p − p dy ≤ Ct (1 − e ) ≤ C x − x . t t Similarly, F(x) − F(y) F(x ) − F(y) (3.4) p − p dy tF (y) tF (y) 2 2 ≤ C F(x) − F(x ) ≤ C x − x , where the constant C depends on F . Then the corresponding part in (3.2)isbounded dt from above by |x − x | ≤ C|x − x |. |x−x | 370 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Step 3.(B): For t ≤ |x − x |, using the Taylor inequality |F(x) − F(y) − F (y)(x − y)|≤ F |x − y| and the mean value inequality (as we have assumed that K is con- vex), x − y F(x) − F(y) (3.5) p − p t tF (y) 2 2 F(x)−F(y) |x − y| |x − y| |x − y| − inf α(x−y)+(1−α) α∈(0,1) F (y) 2t ≤ C + e . t t t Step 3.(B): case (a). If y ∈ B(x,ε) for ε small enough (depending only on F ), we 1 2 have, using again |F(x) − F(y) − F (y)(x − y)|≤ F |x − y| , uniformly in α ∈ (0, 1), F(x) − F(y) 1 1 α(x − y) + (1 − α) ≥|x − y|− F |x − y| ≥ |x − y|. F (y) 2 2 Therefore, for such y’s we have, coming back to (3.5), |x−y| x − y F(x) − F(y) |x − y| 4t p − p ≤ C e . t tF (y) t For this case we get the bound x − y F(x) − F(y) p − p dy t tF (y) B(x,ε) 6 2 |x−y| |x − y| −2 4 2t ≤ C e = Ct E(|B 2 | ) ≤ Ct , B(x,ε) where B denotes a two-dimensional Gaussian variable with covariance matrix t times |x−x | dt 4 the identity. This term contributes to (3.2)as C t ≤ C|x − x |. Step 3.(B): case (b). Now, for t ≤ |x − x | and y ∈ U \ B(x,ε) we write √ √ |x−x | |x−x | dt x − y dt p dy ≤ C P(|B 2 | >ε) t t t 0 U\B(x,ε) 0 |x−x | dt 2t ≤ C e ≤ C x − x , and similarly |x−x | dt F(x) − F(y) p dy ≤ C x − x , t tF (y) 0 U\B(x,ε) −1 where the constant C depends on F and (F ) . U U TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 371 Applying Step 3.(A) and then Step 3.(B) twice (once for x and then again for x )to (δ) (δ) (3.2), we get E((φ (x) − φ (x )) ) ≤ C |x − x |. 1 1 (δ) (δ) | | Second term. We want to prove here that E((φ (x) − φ (x )) ) ≤ C x − x . 2 2 Note that three terms contribute to δφ . The third one is a nice Gaussian ﬁeld indepen- dent of δ. The ﬁrst two terms are similar, so we will just focus on the ﬁrst one, namely (δ) 1 φ (x) := p (x − y)W(dy, dt).Wehave, similarlyto(3.2)and (3.3), c 2 2,1 U δ (δ) (δ) E φ (x) − φ (x ) 2,1 2,1 t t = p (x − y) − p (x − y) dydt 2 2 2 c δ U dt x − y x − y ≤ C p − p dy t c t t 0 U |x−x | dt x − y x − y ≤ C p + p dy + C|x − x |. t c t t 0 U The remaining term can be controlled as follows (noting the symmetry between x and x ): √ √ |x−x | |x−x | |x−y| dt 1 dt 2t e dy ≤ C P(|B 2 | > d) t c t t 0 U 0 |x−x | dt 2t ≤ C e ≤ C x − x , (δ) (δ) c 2 where d = d(K, U ).Thus E((φ (x) − φ (x )) ) ≤ C |x − x |. 2 2 (δ) Third term. We give here a bound on the pointwise variance of φ . By using |x−y| 2 − x−y δ F(x)−F(y) | | (δ) Ct dt e ≥ we get E(φ (x) ) ≤ dy ≤ C. 2 2 F (y) C cδ t R t 3.2. Russo–Seymour–Welsh estimates. — The main result of this section is the follow- ing RSW estimate. It shows that appropriately-chosen quantiles of crossing distances of “long” and “short” rectangles at the same scale can be related by a multiplicative factor that is uniform in the scale. This is the equivalent of Theorem 3.1 from [18] but with the ﬁeld molliﬁed by the heat kernel instead of a compactly-supported kernel. It holds for any ﬁxed ξ> 0. Proposition 7 (RSW estimates for φ ).— If [A, B]⊂ (0, ∞), there exists C > 0 such that a a for (a, b), (a , b ) ∈[A, B] with < 1 < ,for n ≥ 0 and ε< 1/2, we have, b b (n) (n) C log |ε/C| (3.6) (φ, ε/C) ≤ C (φ, ε)e ; a ,b a,b (n) (n) C C log |ε/C| ¯ ¯ (3.7) (φ, 3ε ) ≤ C (φ, ε)e . a ,b a,b 372 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET The following corollary then follows from Propositions 5 and 7. Corollary 8 (RSW estimates for ψ ).— Under the same assumptions as used in Proposition 7, we have (n) (n) C log |ε/C| (3.8) (ψ, ε/C) ≤ C (ψ, ε)e a ,b a,b and (n) (n) C log |ε/C| ¯ ¯ (3.9) (ψ, 3ε ) ≤ C (ψ, ε)e . a,b a ,b We point out that the constants C in (3.8)and (3.9) are not equal to those in (3.6) and (3.7). The remaining parts of the section will only deal with approximations associ- ated with φ so we will omit this dependence in the various observables. We describe below the main lines of the argument. Consider R and R ,two a,b a ,b a a rectangles with respective side lengths (a, b) and (a , b ) satisfying < 1 < . Suppose b b that we could take a conformal map F : R → R mapping the long left and right a,b a ,b sides of R to the short left and right sides of R . (This is not in fact possible since a,b a ,b there are only three degrees of freedom in the choice of a conformal map, but for the sake of illustration we will consider this idealized setting ﬁrst.) Then the proof goes as follows. Take a geodesic π ˜ for φ for the left–right crossing of R . Then, using the cou- 0,n a,b pling (3.1), we have φ φ ξφ (F(π( ˜ t))) 0,n 0,n 0,n L (R ) ≤ L (F(π) ˜ ) = e |F (π( ˜ t))|·|π˜ (t)|dt a ,b ξ(φ +δφ +δφ ) 0,n L H ≤ F e ds a,b π ˜ ξ δφ ˜ L ξ φ ξδφ R 0,n H a,b ≤ F e e e ds. a,b π ˜ ˜ ˜ It is essential that π ˜ is φ measurable and φ is independent of δφ . Then, we can use 0,n 0,n H the following lemma. Lemma 9. — If is a continuous ﬁeld and is an independent continuous centered Gaussian ﬁeld with pointwise variance bounded above by σ > 0, then we have, as long as ε is sufﬁciently small compared to σ , 2 −1 2σ log ε 1. ( + , ε) ≤ e (, 2ε); 1,1 1,1 2 −1 2σ log ε ¯ ¯ 2. ( + , 2ε) ≤ e (, ε). 1,1 1,1 2 −1 Proof. — Fix s := 2σ log ε throughout the proof. Let π() be a geodesic as- sociated with the left–right crossing length for the ﬁeld , and deﬁne the measure μ on TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 373 −1 π() by μ(ds) = L () e ds,so e ds = 1. Conditionally on , using Jensen’s in- 1,1 π() −1 2 equality with α = = (log ε )/(2σ ), which is greater than 1 for small enough ε, 2σ and Chebyshev’s inequality, we have + s α αs (3.10) P e ds > e L () | ≤ P e dμ ≥ e | 1,1 π() π() 1 2 2 α σ −αs 2 2σ ≤ e e = e = ε. To bound from above L ( + ), we take a geodesic for and use the moment estimate 1,1 2 −1 (3.10). We start with the left tail. Still with s := 2σ log ε ,wehave −s P L () ≤ ( + , ε)e 1,1 1,1 s −s ≤ P L ( + ) ≤ e L (), L () ≤ ( + , ε)e 1,1 1,1 1,1 1,1 + P L ( + ) > e L () 1,1 1,1 + s ≤ P L ( + ) ≤ ( + , ε) + P e ds > e L () 1,1 1,1 1,1 π() ≤ 2ε. For the right tail, we have similarly that P L ( + ) ≥ (, ε)e 1,1 1,1 ¯ ¯ ≤ P L ( + ) ≥ (, ε)e , (, ε) ≥ L () 1,1 1,1 1,1 1,1 + P L () ≥ (, ε) 1,1 1,1 ≤ P L ( + ) ≥ e L () + ε ≤ 2ε, 1,1 1,1 which concludes the proof of the lemma. The previous reasoning does not apply directly to rectangle crossing lengths but provides the following proposition. Recall that K is a compact subset of U. Let A,Bbe two boundary arcs of K and denote by L the distance from A to B in K for the metric ξφ 0,n e ds;wedenote A := F(A),B := F(B),K := F(K),and L is the distance from A to ξ φ 0,n B in K for e ds.Recallthatwehave |F |≥ 1 on U. In the application we will achieve this by scaling U to be sufﬁciently small. Proposition 10. — We have the following comparisons between quantiles. There exists C > 0 such that 1. if for some l > 0 and ε< 1/2, P (L ≤ l) ≥ ε,then P (L ≤ l ) ≥ ε/4 with l = C log ε/2C | | F e . 2. if for some l > 0 and ε< 1/2, P (L ≤ l) ≥ 1 − ε, then P (L ≤ l ) ≥ 1 − 3ε with C log ε/2C | | l = F e . K 374 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET FIG.1.—Illustration of Lemma 12 Now, we want to prove a similar result for rectangle crossing lengths. We will need the three following lemmas that were used in [18]. The ﬁrst one is a geometrical con- struction, the second one is a complex analysis result and the last one comes essentially from [39] together with an approximation argument. In these lemmas, by “crossings” we mean continuous path from marked sides to marked sides. Lemma 11 (Lemma 4.8 of [18]).— If a and b are two positive real numbers with a < b, there exists j = j(b/a) and j rectangles isometric to [0, a/2]×[0, b/2] such that if π is a left–right crossing of the rectangle [0, a]×[0, b], at least one of the j rectangles is crossed in the thin direction by a subpath of that crossing. Lemma 12 (Step 1 in the proof of Theorem 3.1 in [18]).— If a/b < 1 and a /b > 1,there exists m, p ≥ 1 and two ellipses E , E with marked arcs (AB), (CD) for E and (A B ), (C D ) for p p E such that: p p 1. Any left–right crossing of [0, a/2 ]×[0, b/2 ] is a crossing of E . 2. Any crossing of E is a left–right crossing of [0, a ]×[0, b ]. 3. When dividing the marked sides of E into m subarcs of equal length, for any pair of such subarcs (one on each side), there exists a conformal map F : E → E and the pair of subarcs is mapped to subarcs of the marked sides of E . 4. For each pair, the associated map F extends to a conformal equivalence U → V where E ⊂ U, E ⊂ V and |F |≥ 1 on U. We refer the reader to Figure 1 for an illustration. Lemma 13 (Positive association and square-root-trick).— If k ≥ 2 and (R ,..., R ) denote 1 k a collection of k rectangles, then, for (x ,..., x ) ∈ (0, ∞) , we have 1 k (n) (n) P L (R )> x ,..., L (R )> x 1 1 k k (n) (n) ≥ P L (R )> x ··· P L (R )> x . 1 1 k k TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 375 An easy consequence of this positive association is the so-called “square-root-trick”: 1/k (n) (n) max P L (R ) ≤ x ≥ 1 − 1 − P ∃i ≤ k : L (R ) ≤ x . i i i i i≤k The main result of this section, Proposition 7, is a rephrasing of the following one. Proposition 14. — We have the following comparisons between quantiles. If a/b < 1 and a /b > 1, there exists C > 0 such that, for any ε ∈ (0, 1/2), (n) (n) C log ε/C | | 1. if P L ≤ l ≥ ε, then P L ≤ Cle ≥ ε/C, a,b a ,b (n) (n) C log ε/C | | 1/C 2. and if P L ≤ l ≥ 1 − ε, then P L ≤ Cle ≥ 1 − 3ε . a,b a ,b Proof. — We provide ﬁrst a comparison between low quantiles and then a compar- ison between high quantiles. (n) Step 1: Comparison of small quantiles. Suppose P(L ≤ l) ≥ ε. By Lemma 11 and a,b (n) (n) union bound, P(L ≤ l) ≥ ε/j . Furthermore, by iterating, we have P(L ≤ l) ≥ p p a/2,b/2 a/2 ,b/2 ε/j . Under this event, by Lemma 12, there exists a crossing of E between two subarcs p 2 of E (one on each side) hence with probability at least ε/(j m ), one of these crossings has length at most l . By the left tail estimate Proposition 10 and Lemma 12,weobtaina C > 0 (depending also on F )suchthatfor all ε, l > 0: p 2 (n) (n) C log ε/(2Cj m ) p 2 | | P L ≤ l ≥ ε ⇒ P L ≤ Cle ≥ ε/(4j m ), a,b a ,b hence the ﬁrst assertion. (n) Step 2: Comparison of high quantiles. Now suppose P(L ≤ l) ≥ 1 − ε.By a,b Lemma 11 (to start with a crossing at a lower scale) and Lemma 13 (square-root-trick), (n) (n) 1/j we have P(L ≤ l) ≥ 1 − ε . Furthermore, by iterating, we have P(L ≤ l) ≥ p p a/2,b/2 a/2 ,b/2 (n) 1/j 1 − ε .Onthe event {L ≤ l}, the ellipse E from Lemma 12 has a crossing of p p a/2 ,b/2 length ≤ l between two marked arcs. Again by subdividing each its marked arcs into m subarcs and applying the square-root trick, we see that for at least one pair of subarcs, −p −2 j m there is a crossing of length ≤ l with probability ≥ 1 − ε . Combining with the right- tail estimate Proposition 10 and Lemma 12,weget: (n) (n) C log ε/C 1/C | | (3.11) P L ≤ l ≥ 1 − ε ⇒ P L ≤ Cle ≥ 1 − 3ε , a,b a ,b which completes the proof. Remark 15. — The importance of the Russo–Seymour–Welsh estimates comes from the following: percolation arguments/estimates work well when taking small quan- tiles associated with short crossings and high quantiles associated with long crossings. Thanks to the RSW estimates, we can instead keep track only of low and high quantiles associated to the unit square crossing, (p) and (p). n n 376 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET 4. Tail estimates with respect to ﬁxed quantiles Lower tails. — This is where we take r small enough (recall the deﬁnition (2.11)) to obtain some small range of dependence of the ﬁeld ψ so that a Fernique-type argument works. Proposition 16 (Lower tail estimates for ψ ).— We have the following lower tail estimate: for p small enough, but ﬁxed, there is a constant C so that for all s > 0, (n) −s −cs (4.1) P L (ψ ) ≤ e (ψ, p) ≤ Ce . 1,3 Proof. — The RSW estimate (3.8)gives (n) (n) −1 −Cξ | log Cε| (4.2) P L (ψ ) ≤ l ≤ ε ⇒ P L (ψ ) ≤ lC e ≤ Cε 3,3 1,3 (n) Now, if L (ψ ) is less than l,thenboth [0, 1]×[0, 3] and [2, 3]×[0, 3] have a left– 3,3 right crossing of length ≤ l and the restrictions of the ﬁeld to these two rectangles are independent (if r deﬁned in (2.11) is small enough). Consequently, (n) (n) (4.3) P L (ψ ) ≤ l ≤ P L (ψ ) ≤ l 3,3 1,3 (n) Take p small, such that C p < 1 where C is the constant in (4.2) and set r := 0 0 (n) (ψ, p ). (This is not related to r , deﬁned previously.) For i ≥ 0, set 0 0 3,3 (4.4) p := (Cp ) i+1 i (n) (n) −1 (4.5) r := r C exp(−Cξ | log(Cp )|) i+1 i By induction we get, for i ≥ 0, (n) (n) (4.6) P(L (ψ ) ≤ r ) ≤ p 3,3 i Indeed, the case i = 0 follows by deﬁnition and then notice that the RSW estimate (4.2) (n) (n) (n) (n) under the induction hypothesis implies that P(L (ψ ) ≤ r ) ≤ p ⇒ P(L (ψ ) ≤ r ) ≤ 3,3 i 1,3 i+1 (n) (n) (n) (n) 2 2 Cp which gives, using (4.3), P(L (ψ ) ≤ r ) ≤ P(L (ψ ) ≤ r ) ≤ (Cp ) = p . i i i+1 3,3 i+1 1,3 i+1 2 2 −2 From (4.4)weget p = (p C ) C and from (4.5) we have the lower bound, for i 0 i ≥ 1, √ √ i−1 2 i/2 (n) (n) −i −Cξ | log(Cp )| (n) −Ci −Cξ | log p C |2 k 0 k=0 r ≥ (ψ, p )C e ≥ (ψ, p )e e . 0 0 i 3,3 3,3 Our estimate (4.6) then takes the form, for i ≥ 0, 2 i/2 (n) (n) −Ci −ξ C | log p C |2 2 −2 P L (ψ ) ≤ (ψ, p )e e ≤ p C C . 0 0 3,3 3,3 TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 377 This can be rewritten, taking i =2log s,as (n) (n) −1 −Clog s −ξ s −cs P L (ψ ) ≤ (ψ, p )C e e ≤ e 3,3 3,3 for s > 2 with absolute constants. We obtain the statement of the proposition by using again the RSW estimates. Using the comparison result between φ and ψ (Proposition 5), we get the following corollary. Corollary 17 (Lower tail estimates for φ).— For p small enough, but ﬁxed, for all s > 0 we have a constant C < ∞ so that (n) −s −cs (4.7) P L (φ) ≤ e (φ, p) ≤ Ce . 1,3 Upper tails. — The proof for the upper tails is similar to the one of Proposition 5.3 in [18]. Themaindifferenceisthatwehavetoswitchbetween φ and ψ,sothatwecan use the independence properties of ψ together with the scaling properties of φ.Before stating the proposition, we refer the reader to (2.20) for the deﬁnition of (φ, p).In contrast with the lower tails estimates which are relative to (φ, p),wedonot know how to prove (at least a priori) the analogous result for the upper tails with (φ, p) only. However, we can prove it by replacing (φ, p) by (φ, p) (φ, p) and this is the content n n n of the following proposition. Proposition 18 (Upper tail estimates for φ).— For p small enough, but ﬁxed, we have a constant C < ∞ so that for all n ≥ 0 and s > 2, (n) s c log s (4.8) P L (φ) ≥ e (φ, p) (φ, p) ≤ Ce . n n 3,1 Proof. — The proof uses percolation and scaling arguments. A percolation argu- ment is used to build a crossing of a larger rectangle from smaller annular circuits, and then a scaling argument is used to relate quantiles of these annular crossings to crossing quantiles of the larger rectangle. Step 1: Percolation argument. To each unit square P of Z , we associate the four crossings of long rectangles of size (3, 1) surrounding P, each comprising three squares on one side of the eight-square annulus surrounding P, as illustrated in Figure 2.Wedeﬁne (n) S (P,ψ) to be the sum of the four crossing lengths, and declare the site P to be open (n) (n) when the event {S (ψ, P) ≤ 4 (ψ, p)} occurs. This occurs with probability at least 1 − 3,1 (n) (n) ε(p),where ε(p) goes to zero as p goes to zero (recall that P(L (ψ ) ≤ (p)) = 1 − p). 3,1 3,1 Using a highly supercritical ﬁnite-range site percolation estimate to obtain exponential decay of the probability of a left–right crossing (which is standard technique in classical 378 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET FIG. 2. — Four blue rectangles are surrounding the square P. Left–right geodesics associated to the long and short rectangles surrounding P are drawn in green and brown respectively. Any geodesic π , here in red, which intersects P has to cross the green circuit and to induce a short crossing of one of the four rectangles (Color ﬁgure online) percolation theory [20]; see also for example the proof of Proposition 4.2 in [10]) together with the Russo–Seymour–Welsh estimates (to come back to (ψ, p)), we have (n) 2 −ck P L (ψ ) ≥ Ck (ψ, p) ≤ Ce . 3k,k Therefore, using this bound together with Proposition 5 to bound X (recalling the 3k,k deﬁnition (2.18)), (n) ξ C k 2 P L (φ) ≥ e C Ck (φ, p/2) p n 3k,k (n) ξ X ξ C k 2 3k,k ≤ P e L (ψ ) ≥ e C Ck (φ, p/2) p n 3k,k (n) ≤ P X ≥ C k + P L (ψ ) ≥ C Ck (φ, p/2) 3k,k p n 3k,k (n) −ck 2 −ck ≤ Ce + P L (ψ ) ≥ Ck (ψ, p) ≤ Ce . 3k,k ¯ ¯ Note that we used the bound (ψ, p) ≤ C (φ, p/2) from (2.22) in the third inequality; n p n here C is deﬁned as in (2.22). Step 2: Decoupling and scaling. In this step, we give a rough bound of the coarse ﬁeld φ , to obtain spatial independence of the remaining ﬁeld between blocks of size 0,m −m 2 . When an event occurs on one block with high enough probability, the percolation argument of Step 1 then provides, with very high probability, a left–right path of such (n) ξ max φ (m,n) R 0,m 3,1 events occurring simultaneously. Since L (φ) ≤ e L (φ), the scaling prop- 3,1 3,1 (d) (m,n) (n−m) −m erty of the ﬁeld φ, i.e. L (φ) = 2 L m m (φ),gives 3,1 3·2 ,2 (n) ξ s m c 2 P L (φ) ≥ e e (φ, p) n−m 3,1 (n−m) −m c 2 ≤ P max φ ≥ Cm + s m + P 2 L (φ) ≥ e (φ, p) m m 0,m n−m 3·2 ,2 3,1 2 m −cs −c2 ≤ Ce + Ce , TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 379 −1/2 where the ﬁrst term of the second expression is bounded by taking a = C + sm in Proposition 2 and the second bound follows from the result obtained in Step 1 with m Cm k = 2 , taking a slightly larger c in exp(c 2 ) to absorb the factor e . −2ξ k −C k Step 3: We derive an a priori bound (φ, p) ≥ 2 (φ, p)e . (Note that the n n−k −k argument below will be optimized in (5.31).) For each dyadic block of size 2 visited by −k π (φ), one of the four rectangles of size 2 (1, 3) around P has to be crossed by π (φ). n n k −k Therefore, since π (φ) has to visit at least 2 dyadic blocks of size 2 ,wehave (n) k ξ inf φ (k,n) S 2 0,k [0,1] L (φ) ≥ 2 e min min L (R (P), φ), 1,1 i P∈P ,P∩π (φ)=∅ 1≤i≤4 k n S −k where (R (P)) denote the four long rectangles of size 2 (1, 3) surrounding P. Us- 1≤i≤4 ing the supremum tail estimate (2.3) and the left tail estimates (4.7), we get (φ, p) ≥ −2ξ k −C k 2 (φ, p)e . Indeed, n−k ξ inf φ k (k,n) S −2ξ k −C k 2 0,k [0,1] P e min min 2 L (R (P), φ)≤2 (φ, p)e n−k 1≤i≤4 P∈P ,P∩π (φ)=∅ k n ≤ P inf φ ≤−k log 4 − C k 0,k [0,1] k (k,n) S −C k + P min min 2 L (R (P), φ) ≤ (φ, p)e n−k P∈P ,P∩π (φ)=∅ 1≤i≤4 k n and each term is less than p/2 if C is large enough, depending on p. Therefore, we have 2ξ m C m (φ, p) ≤ (φ, p) (φ, p) ≤ 2 e (φ, p) (φ, p). n−m n−m n−m n−m n 2 m Now, by coming back to the partial result obtained in Step 2 and by taking s = 2 for n/2 s ∈[1, 2 ],weget (n) cs log s cs −cs P L (φ) ≥ e e (φ, p) (φ, p) ≤ e . n n 3,1 Step 4: Now we consider large tails, so we assume s ≥ 2 . By a direct comparison −ξ(2n+C n) with the supremum, we have (φ, p) ≥ 2 (lateronwewilluse amoreprecise estimate from [12], see (5.5)). Moreover, bounding from above the left–right distance by taking a straight path from left to right and then using a moment method analogous to (n) ξ s 2(n+1) log 2 the one in (3.10), we get P L (φ) ≥ e ≤ e . Altogether, 1,1 (n) (n) ξ s ξ s P L (φ) ≥ (φ, p) (φ, p)e ≤ P L (φ) ≥ (φ, p)e n n n 1,1 1,1 (s−n log 4−C n) − Cs −c 2(n+1) log 2 log s ≤ e ≤ e e , 380 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET −ξ(2n+C n) where we used (φ, p) ≥ 1 in the ﬁrst inequality and the bound (φ, p) ≥ 2 n n (n) − ξ s 2(n+1) log 2 together with the tail estimate P L (φ) ≥ e ≤ e in the second one. The last 1,1 inequality follow since s ≥ 2 . n/2 Combining the tail estimate of Step 3, valid for s ∈[1, 2 ], and the one of Step 4, n/2 valid for s ≥ 2 , completes the proof. Using again the comparison between φ and ψ given in Proposition 5,weget the following corollary. Corollary 19 (Upper tail estimates for ψ ).— For p small enough, but ﬁxed, we have, for all n ≥ 0 and s > 2, (n) s c log s (4.9) P L (ψ ) ≥ e (ψ, p) (ψ, p) ≤ Ce . n n 3,1 5. Concentration 5.1. Concentration of the log of the left–right crossing length. Condition (T). — Denote by π (ψ ) the left–right geodesic of the unit square associated to the ﬁeld ψ . If there are multiple such geodesics, let π (ψ ) be chosen among them 0,n n in some measurable way, for example by taking the uppermost geodesic. By π (ψ ) its K-coarse graining which we deﬁne as (5.1) π (ψ ) := {P ∈ P : P ∩ π (ψ ) = ∅}, K n recalling the deﬁnition (2.7)of P .Let ψ (P) denote the value of the ﬁeld ψ taken at K 0,n 0,n the center of a block P. We introduce the following condition: there exist constants α> 1, c > 0 so that for K large we have ⎛ ⎛ ⎞ ⎞ 1/α 2ξψ (P) 0,K P∈π (ψ ) ⎜ ⎜ ⎟ ⎟ n −cK (5.2)sup E ≤ e . (Condition (T)) ⎝ ⎝ ⎠ ⎠ n≥K ξψ (P) 0,K P∈π (ψ ) The importance of Condition (T) comes from the following theorem. (n) Theorem 20. — If ξ is such that Condition (T) above is satisﬁed, then (log L (φ) − 1,1 (n) log λ (φ)) is tight, where λ (φ) denotes the median of L . n n≥0 n 1,1 It is not expected that the weight is approximately constant over the crossing (since there may be some large level lines of the ﬁeld that the crossing must cross). Condition (T), however, roughly requires that the length of the crossing is supported by a number of TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 381 coarse blocks that grows at least like some small but positive power of the total number of coarse blocks. Note that the fraction in Condition (T) is the norm of the vector of crossing weights on each block divided by the square of the norm of the same, and thus controlling it amounts to an anticoncentration condition for this vector. The core of this section is the proof of Theorem 20. Before proving it, let us al- ready jump to the important following proposition. Here we use the assumption that ξ ∈ (0, 2/d ), although the formulation of Condition (T) is designed so that it could also hold for larger ξ . Proposition 21. — If γ ∈ (0, 2), then ξ := satisﬁes Condition (T). Proof. — Step 1: Supremum bound. Taking the supremum over all blocks of size −K 2 2 in [0, 1] ,weget 2ξψ (P) 0,K ξ max ψ (P) ξ max φ (P) P∈P 0,K P∈P 0,K K K K e e P∈π (ψ ) n ξ X ≤ ≤ e , ξψ (P) ξψ (P) 0,K 0,K e e K K ξψ (P) P∈π (ψ ) P∈π (ψ ) 0,K n n P∈π (ψ ) recalling the deﬁnition of X below (2.18). Step 2: We give a lower bound of the denominator of the right-hand side. By taking the concatenation of straight paths in each box of π (ψ ), we get a left–right crossing of [0, 1] . Denote this crossing by .Wehave, n,K,ψ ξψ (P) −ξ X ξφ (P) 0,K 1 0,K (5.3) e ≥ e e K K P∈π (ψ ) P∈π (ψ ) n n −ξ X K (K) ≥ e exp(−ξ max osc (φ ))2 L (φ, ) P 0,K n,K,ψ P∈P (K) −ξ X K ≥ e exp(−ξ max osc (φ ))2 L (φ), P 0,K 1,1 P∈P where osc was deﬁned in (2.6)and P was deﬁned in (2.8). Step 3: Combining the two previous steps, we have ξ max φ (P) 1 0,K 2ξψ (P) 0,K P∈P K K e ξ max osc (φ ) P∈π (ψ ) 1 P 0,K n 2ξ X 1 P∈P ≤ e e . 2 (K) 2 L (φ) ξψ (P) 0,K 1,1 P∈π (ψ ) 1 1 Now, we take α> 1 close to 1. Using Hölder’s inequality with + = 1and r close to 1, r s together with Cauchy–Schwarz, we get ⎛ ⎛ ⎞ ⎞ 1/α 2ξψ (P) 0,K P∈π (ψ ) ⎜ ⎜ ⎟ ⎟ ⎝ ⎝ ⎠ ⎠ ξψ (P) 0,K P∈π (ψ ) n 382 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET 1/α αξ max φ (P) P∈P 0,K e αξ max osc (φ ) 1 P 0,K −K 2αξ X 1 P∈P ≤ 2 E e e (K) (L (φ)) 1,1 1/2αs 1/αr −2αs αrξ max φ (P) 1 0,K −K (K) P∈P ≤ 2 E e E L (φ) 1,1 1/4αs 4αsξ max osc (φ ) 1/4αs 1 P 0,K 8αsξ X 1 P∈P × E e E e . Therefore, using (2.4)for themaximum,(4.7) for the left–right crossing, Proposition 5 to bound X and (2.10) for the maximum of oscillations, we ﬁnally get, when αrξ< 2(recall that αr can be taken arbitrarily close to 1), ⎛ ⎛ ⎞ ⎞ α 1/α 2ξψ (P) 0,K ⎜ ⎜ P∈π (ψ ) ⎟ ⎟ (K) n −K 2ξ K −1 C K (5.4) E ≤ 2 2 (φ, p) e . ⎝ ⎝ ⎠ ⎠ 1,1 ξψ (P) 0,K P∈π (ψ ) 2 γ Step 4: Lower bound on quantiles. For γ ∈ (0, 2),Q := + > 2. Using Propo- γ 2 sition 3.17 from [12] (circle average LFPP) and Proposition 3.3 from [11](comparison between φ and circle average), we have, if p is ﬁxed and ε ∈ (0, Q − 2), for K large enough, (K) −K(1−ξ Q+ξε) (5.5) (φ, p) ≥ 2 . 1,1 Step 5: Conclusion. Using the results from the two previous steps, we ﬁnally get ⎛ ⎛ ⎞ ⎞ α 1/α 2ξψ (P) 0,K ⎜ ⎜ P∈π (ψ ) ⎟ ⎟ n −ξ(Q−2−ε)K C K E ≤ 2 e , ⎝ ⎝ ⎠ ⎠ ξψ (P) 0,K P∈π (ψ ) which completes the proof. Now, we come back to the proof of Theorem 20. We ﬁrst derive a priori estimates on the quantile ratios. Lemma 22. — Let Z be a random variable with ﬁnite variance and p ∈ (0, 1/2). If a pair ¯ ¯ ¯ ((Z, p), (Z, p)) satisﬁes (Z, p) ≥ (Z, p), P(Z ≥ (Z, p)) ≥ pand P(Z ≤ (Z, p)) ≥ p, then, we have: (5.6) ((Z, p) − (Z, p)) ≤ Var Z. Proof. — If Z is an independent copy of Z, notice that for l ≥ l we have 2Var(Z) = 2 2 2 E((Z − Z) ) ≥ E(1 1 (Z − Z) ) ≥ P(Z ≥ l )P(Z ≤ l)(l − l) . Z ≥l Z≤l TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 383 (n) In the following lemma, we derive an a priori bound on the variance of log L (φ). 1,1 Lemma 23. — For all n ≥ 0 we have the bound (n) Var log L (φ) ≤ ξ (n + 1) log 2 1,1 (n) Proof. — Denote by L (D ) the left–right distance of [0, 1] for the length metric 1,1 ξφ k −k 0,n e ds,where φ is piecewise constant on each dyadic block of size 2 where it is equal 0,n to the value of φ at the center of this block. (We do not assign an independent meaning 0,n to the notation D .) Note that we have −k −k −C2 ∇φ (n) (n) (n) C2 ∇φ 0,n 0,n 2 2 [0,1] [0,1] e L ≤ L (D ) ≤ L e , 1,1 1,1 1,1 (n) (n) which gives almost surely that L (φ) = lim L (D ). By dominated convergence we k→∞ k 1,1 1,1 have (n) (n) Var log L (φ) = lim Var log L (D ). 1,1 1,1 k→∞ (n) Now, log L (D ) is a ξ -Lipschitz function of p = 4 Gaussian variables denoted by 1,1 Y = (Y , ..., Y ),where on R we use the supremum metric. We can write Y = AN for 1 p some symmetric positive semideﬁnite matrix A and standard Gaussian vector N on R . (n) Then log L (D ) = f (Y) = f (AN) which is ξσ -Lipschitz as a function of N where 1,1 σ = max(|A |, ..., |A |). By the Gaussian concentration inequality of [17, Lemma 2.1], 1 p applied as in [10, Lemma 5.8], since the pointwise variance of the ﬁeld is (n + 1) log 2 we have (n) Var log L (D ) ≤ max(Var(Y ), ..., Var(Y )) = ξ (n + 1) log 2. k 1 p 1,1 Before stating the following lemma, we refer the reader to the deﬁnition of quantile ratios in (2.20). Lemma 24 (A priori bound on the quantile ratios).— Fix p ∈ (0, 1/2). There exists a constant C depending only on p such that for all n ≥ 1, C n (5.7) (ψ, p) ≤ e . (k) Proof. — By using Lemma 23 we get Var(log L (ψ )) ≤ Ck for all 1 ≤ k ≤ n and 1,1 an absolute constant C > 0. This implies the same bound for ψ by Proposition 5. Using (k) (ψ,p) then Lemma 22 with Z = log L (ψ ) for k ≤ n, we ﬁnally get the bound max ≤ k k≤n 1,1 (ψ,p) C n e . 384 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Proof of Theorem 20.— The proof is divided in ﬁve steps. K will denote a large pos- itive number to be ﬁxed at the last step. Step 1. Quantiles-variance relation/setup. We aim to get an inductive bound on (ψ,p/2) (ψ, p). We will therefore bound in term of ’s at lower scales. p will be ﬁxed (ψ,p/2) from now on, small enough so that we have the tail estimates from Section 4 for φ with p and for ψ with p/2. The starting point is the bound (n) (ψ, p/2) C Var log L (ψ ) 1,1 (5.8) ≤ e . (ψ, p/2) Step 2. Efron–Stein. Using the Efron–Stein inequality with the block decomposition of ψ introduced in (2.13), deﬁning the length with respect to the unresampled ﬁeld 0,n (n) L (ψ ) = L (ψ ),weget 1,1 (n) (5.9) Var log L (ψ ) ≤ E log L (ψ ) − log L (ψ ) 1,1 n + E log L (ψ ) − log L (ψ ) , P∈P where in the ﬁrst term (resp. second term) we resample the ﬁeld ψ (resp. ψ )toget 0,K K,n,P ˜ ˜ an independent copy ψ (resp. ψ ) and we consider the left–right distance L (ψ ) 0,K K,n,P (resp. L (ψ )) of the unit square associated to the ﬁeld ψ − ψ + ψ (resp. ψ − 0,n 0,K 0,K 0,n ψ + ψ ). K,n,P K,n,P Step 3. Analysis of the ﬁrst term. For the ﬁrst term, using Gaussian concentration as in the proof of Lemma 23,weget K 2 (5.10) E((log L (ψ ) − log L (ψ )) ) = 2E(Var(log L (ψ )|ψ − ψ )) ≤ CK. n n 0,n 0,K Step 4. Analysis of the second term. For P ∈ P ,if L (ψ ) > L (ψ ), the blockPis K n visited by the geodesic π (ψ ) associated to L (ψ ).Deﬁne n n K ε −K (5.11)P := {Q ∈ P : d(P, Q) ≤ CK 2 }, where we recall that ε is associated with the range of dependence of the resampled ﬁeld ψ through (2.11) (see also the subsection following this deﬁnition). Here, d(P, Q) is K,n,P the L -distance between the sets P and Q. We upper-bound L (ψ ) by taking the concatenation of the part of π (ψ ) outside of P together with four geodesics associated to long crossings in rectangles comprising a K P K circuit around P (for the ﬁeld ψ which coincides with the ﬁeld ψ outside of P ). We 0,n 0,n −K ε K K get, introducing the rectangles (Q (P)) of size 2 (CK , 3) surrounding P (P and i 1≤i≤4 −K its 3 · 2 neighborhood form an annulus, and gluing the four crossings gives a circuit in this annulus) and using the inequality log x ≤ x − 1, P (n) (L (ψ ) − L (ψ )) max L (Q (P), ψ ) n + 1≤i≤4 i (5.12) log L (ψ ) − log L (ψ ) ≤ ≤ 4 . + (n) (n) L (ψ ) L (ψ ) 1,1 1,1 TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 385 FIG. 3. — Illustration of the geodesics used in the upper bound of Step 4 • We recall the notation φ (P) to denote the value of the ﬁeld φ at the center 0,K 0,K of P. We bound from above each term in the maximum of (5.12) as follows: (n) ξ X (n) L (Q (P), ψ ) ≤ e L (Q (P), φ) i i ξ X ξφ (P) ξ osc (φ ) (K,n) 0,K K 0,K ≤ e e e L (Q (P), φ) 2ξ X ξψ (P) ξ osc (φ ) (K,n) 0,K K 0,K ≤ e e e L (Q (P), φ), where the oscillation osc is deﬁned in (2.6)and P is deﬁned in (5.11). −K −K 2 L For a rectangle Q of size 2 , with corners in 2 Z ,wedenoteby (R (Q)) 1≤i≤4 −K the four long rectangles of size 2 (3, 1) surrounding Q. We can upper-bound the rect- angle crossing lengths associated to the Q (P)’s by gluing O(K ) rectangle crossings of −K −K size 2 (3, 1), which include an annulus around each block Q of size 2 (1, 1) (with −K 2 K corners in 2 Z ) in the shaded region A of Figure 3.Weget (K,n) ε (K,n) L max L (Q (P), φ) ≤ CK max L (R (Q), φ) 1≤i≤4 Q∈A ,1≤i≤4 and we end up with the following upper bound: (5.13) log L (ψ ) − log L (ψ ) ξψ (P) 0,K 2ξ X ξ osc (φ ) ε (K,n) L K 0,K 0 ≤ e e CK max L (R (Q), φ). (n) Q∈A ,1≤i≤4 L (ψ ) 1,1 • We lower-bound the denominator of (5.13) as follows. If P ∈ P is visited by a π (ψ ) geodesic, then there are at least two short disjoint rectangle crossings among the four surrounding P. Therefore, if we denote by P the box containing P at its center whose size is three times that of P, 386 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET ξψ (n) S −ξ X (n) S 0,n e ds ≥ 2min L (R (P), ψ ) ≥ e min L (R (P), φ) i i 1≤i≤4 1≤i≤4 π (ψ )∩P −ξ X ξφ (P) −ξ osc (φ ) (K,n) S 0,K 0,K ≥ e e e min L (R (P), φ) 1≤i≤4 −2ξ X ξψ (P) −ξ osc (φ ) (K,n) S 0,K ˆ 0,K ≥ e e e min L (R (P), φ), 1≤i≤4 S −K where (R (P)) denote the four short rectangles of size 2 (1, 3) surrounding P. 1≤i≤4 Summing over all P’s and taking uniform bounds for the rectangle crossings at higher scales, (n) ξψ ξψ 0,n 0,n L (ψ ) = e ds ≥ e ds 1,1 P∩π (ψ ) P∩π (ψ ) n n P∈P P∈P K K −2ξ X (K,n) S ≥ e min min L (R (P), φ) 1≤i≤4 P∈P ⎛ ⎞ ξψ (P) −ξ osc (φ ) 0,K ˆ 0,K ⎝ P ⎠ × e e . P∈P ,P∩π (ψ )=∅ K n Therefore, taking a uniform bound for the oscillation, we get ⎛ ⎞ (n) −2ξ X ξψ (P) −ξ osc (φ ) (K,n) S 0,K 0,K ⎝ ˆ ⎠ (5.14) L (ψ ) ≥ e e e min L (R (P), φ) 1,1 i P∈P ,1≤i≤4 P∈π (ψ ) 1 −ξ max osc (φ ) 1 0,K −2ξ X P∈P P (K,n) S (5.15) ≥ e e min L (R (P), φ) 9 P∈P ,1≤i≤4 ξψ (P) 0,K × e . P∈π (ψ ) L −K • We recall that (R (P)) denote the four rectangles of size 2 (3, 1) surround- 1≤i≤4 ing P. Gathering inequalities (5.13)and (5.15), we have E log L (ψ ) − log L (ψ ) P∈P 2ξψ (P) 0,K (K,n) L e 1 max L (R (P), φ) P∈P ,1≤i≤4 ⎜ P∈π (ψ ) i 2ε n K ≤ CK E 2 S (K,n) min 1 L (R (P), φ) P∈P ,1≤i≤4 ξψ (P) i 0,K P∈π (ψ ) Cξ max osc (φ ) 1 K 0,K ⎟ P 8ξ X P∈P × e e . ⎠ TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 387 • Condition (T) gives us a α> 1and c > 0 so that for K large enough, for n ≥ K, ⎛ ⎛ ⎞ ⎞ 1/α 2ξψ (P) 0,K P∈π (ψ ) ⎜ ⎜ ⎟ ⎟ n −cK E ≤ e . ⎝ ⎝ ⎠ ⎠ ξψ (P) 0,K P∈π (ψ ) Then, by using the gradient estimate (2.10) and recalling the deﬁnition of P in (5.11), we have ε −K +ε Cmax osc (φ ) 1 K 0,K 0 0 CK 2 ∇φ 2 P∈P P 0,K CK [0,1] (5.16) E e ≤ E e ≤ e . It is for the second inequality that in (2.11)wetake ε to be small in the deﬁnition of ψ ; ε < 1/2 is sufﬁcient. Furthermore, using our tail estimates with regard to upper and lower quantiles for φ (see (4.7)and (4.8), and the scaling property (2.23), for β> 1so that 1 1 + = 1, we get α β ⎛ ⎞ 1 2β (K,n) L max 1 L (R (P), φ) +ε P∈P ,1≤i≤4 i 0 K 2 CK ⎝ ⎠ (5.17) E ≤ (φ, p)e . n−K (K,n) min 1 L (R (P), φ) P∈P ,1≤i≤4 Note that we could have a log K term instead of the K in (5.17). Altogether, by applying Hölder inequality and Cauchy–Schwarz, we get +ε 2 0 P −cK CK 2 (5.18) E log L (ψ ) − log L (ψ ) ≤ e e (φ, p) n n−K P∈P +ε −cK CK 2 ≤ e e C (ψ, p/2), n−K 2 2 where we used (2.22) in the last inequality to get (φ, p) ≤ C (ψ, p/2). n−K n−K Step 5. Conclusion. Gathering the bounds obtained in Step 3 (inequality (5.10)) and Step 4 (inequality (5.18)), we get, coming back to the inequality (5.9), for K large enough, (n) −C K 2 (5.19)VarlogL (ψ ) ≤ C K + e (ψ, p/2). 1,1 n−K Now, we will show that this bound together with the a priori bound on the quantile ratios (Lemma 24) is enough to conclude ﬁrst that (ψ, p/2)< ∞ and then that (n) sup Var log L (ψ ) < ∞, using the tail estimates (4.7)and (4.9). n≥0 1,1 Coming back to Step 1 (equation (5.8)) and using (5.19), we get the inductive in- equality (5.20) below for K large enough and n ≥ K, and (5.21) below by the a priori bound on the quantile ratios Lemma 24: (n) (ψ, p/2) 2 −C K n 2 C Var log L (ψ ) C C K+e (ψ,p/2) 1,1 p 1 n−K (5.20) ≤ e ≤ e ; (ψ, p/2) C K (5.21) (ψ, p/2) ≤ e . K 388 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET From now on, we take K large enough but ﬁxed so that √ √ −C K C K C 2C K 2 2 p p 1 (5.22) e (e + e ) ≤ C K. Set C 2C K p 1 (5.23) := (ψ, p/2) ∨ e , Rec K so that (ψ, p/2) ≤ . This is the initialization of the induction. Now, assume that K Rec (ψ, p/2) ≤ . In particular, (ψ, p/2) ≤ and using (5.20) n−1 Rec n−K Rec (ψ, p/2) −C K 2 C C K+e p 1 Rec ≤ e (ψ, p/2) C 2C K p 1 The right-hand side is smaller than e and therefore than . Indeed, by (5.23), Rec (5.21)and (5.22), −C K 2 −C K C 2C K 2 2 2 p 1 e ≤ e ( (ψ, p/2) + e ) Rec −C K C K C 2C K 2 2 p p 1 ≤ e (e + e ) ≤ C K. Therefore, (ψ, p/2) (ψ, p/2) = (ψ, p/2) ∨ ≤ . n n−1 Rec (ψ, p/2) Therefore, (ψ, p/2)< ∞ thus (φ, p)< ∞ and by the tail estimates (4.7)and (4.8), ∞ ∞ (n) the sequence (log L (φ) − log λ (φ)) is tight. n n≥0 1,1 5.2. Weak multiplicativity of the characteristic length and error bounds. — Henceforth, we will only consider the case ξ = for γ ∈ (0, 2) and the ﬁeld φ . All observables will 0,n be assumed to be taken with respect to φ and we will drop the additional notation used to differ between φ and ψ . In this case, we saw that there exists a ﬁxed constant C > 0 (n) (n) (n) (n) −1 ¯ ¯ so that for all n ≥ 0, (p) ≤ C (p),C (p) ≤ (p) and with the tail estimates, 3,1 1,3 3,1 1,3 (n) (n) E(L ) ≤ CE(L ). All these characteristic lengths are uniformly comparable. We will 3,1 1,3 (n) take λ to denote one of them, say the median of L . 1,1 In the next elementary lemma, we prove that a sequence satisfying a certain quan- titative weak multiplicative property has an exponent, and we quantify the error. Lemma 25. — Consider a sequence of positive real numbers (λ ) . If there exists C > 0 such n n≥1 that for all n ≥ 1,k ≥ 1 we have √ √ −C k C k (5.24) e λ λ ≤ λ ≤ e λ λ , n k n+k n k n+O( n) then there exists ρ> 0 such that λ = ρ . n TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 389 Proof. — We introduce the sequence (a ) such that λ n+1 = (λ n ) e . By iterat- n n≥0 2 2 ing, we get n+1 n n−1 2 a 4 2a +a 2 2 a +2 a +···+2a +a n n−1 n 0 1 n−1 n n+1 n n−1 λ = (λ ) e = (λ ) e = ··· = λ e . 2 2 2 −n/2 The condition (5.24) gives that the sequence 2 a is bounded, therefore the series n≥0 a a −k/2 −n/2 k k converges and | |≤ 2 (sup 2 |a |) 2 . In particular there exists k k k k≥0 k≥n k≥0 2 2 ρ> 0such that a a a 1 n 1 ∞ n+1 k n+1 k n k n n/2 2 log λ + 2 log λ + −2 1 1 2 O(2 ) 2 k=0 k 2 k=0 k k≥n+1 k 2 2 2 n+1 λ = e = e e = ρ e . Now that we have the existence of an exponent, we prove the upper bound of Lemma 25. There exist C , C > 0 such that we have the following upper bounds: 1 2 k k/2 2 C 2 (5.25) λ k ≤ ρ e , C k (5.26) λ ≤ λ λ e . n+k n k 2 2 C Take C large enough so that (C + C ) + (C + C )C ≤ C and λ ≤ ρe .Wewantto 3 1 2 1 2 3 1 n C n prove by induction that for all n ≥ 1, λ ≤ ρ e . The assumption on C implies that this n 3 k k+1 holds for n = 1. By induction (in a dyadic fashion), take n ∈[2 , 2 ). We decompose n as k k n = 2 + n with n ∈[0, 2 ). We have, by using (5.26), (5.25) and the induction hypothesis, k k k/2 k k/2 k/2 C 2 2 C 2 n C n C 2 2 1 k 3 k 2 λ ≤ λ k λ e ≤ (ρ e )(ρ e )e n 2 n √ √ k/2 n (C +C )2 +C n n C n 1 2 3 k 3 = ρ e ≤ ρ e , since by the assumption on C we have √ √ k/2 2 k k/2 2 (C + C )2 + C n = (C + C ) 2 + (C + C )C 2 n + C n 1 2 3 k 1 2 1 2 3 k k 2 k 2 ≤ C (2 + n ) = C n. 3 3 The proof of the lower bound is similar. In the next proposition we prove that the characteristic length λ satisﬁes the weak multiplicativity property (5.24) and we identify the exponent by using the results of [12]. Proposition 26. — For ξ satisfying Condition (T), there exists C > 0 such that for all n ≥ 1, k ≥ 1 we have √ √ −C k C k (5.27) e λ λ ≤ λ ≤ e λ λ . n k n+k n k Furthermore, when γ ∈ (0, 2) and ξ = γ/d , we have −n(1−ξ Q)+O( n) (5.28) λ = 2 . n 390 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Proof. — Let us assume ﬁrst that (5.27) holds. Then, by using Lemma 25,there n+O( n) exists ρ> 0such that we have λ = ρ . Similarly to (5.5), for each ﬁxed small δ> 0, for k large enough we have, −k(1−ξ Q−δ) (5.29) λ ≤ 2 . The proof of (5.29)follows thesamelinesasthe oneof(5.5). Combining (5.29)and (5.5) −(1−ξ Q) we get ρ = 2 . Now, we prove that the characteristic length satisﬁes (5.27). (k) (k) Step 1: Weak submultiplicativity. Let π be such that L (π ) = L .If P ∈ P is vis- k k k 1,1 (k,n+k) ξφ k,n+k ited by π , consider the concatenation S (P) of four geodesics for e ds associated −k to the rectangles of size 2 (3, 1) surrounding P. Each geodesic is in the long direction (k,n+k) (k,n+k) of its rectangle so that this concatenation is a circuit. By scaling, E(L (S (P))) = (n) −k+2 k 2 E(L ). Note that the collection π (φ) ={P ∈ P : P ∩ π = ∅} is measurable with k k 3,1 k (k,n+k) respect to φ , which is independent of φ .Set := k S (P).Notethat 0,k k,n+k k,n P∈π (φ) contains a left–right crossing of [0, 1] whoselengthisbounded aboveby k,n (n+k) (n+k) (k,n+k) L ( ) = L (S (P)) k,n P∈π (φ) (k,n+k) (k,n+k) ξφ (P) ξ osc (φ ) 0,k 0,k ≤ L (S (P))e e , P∈π (φ) where P denotes the box containing P at its center whose side length is three times that (n+k) (n+k) of P. Since L ≤ L ( ), by independence we have k,n 1,1 ⎛ ⎞ (n+k) (n) −k ξφ (P) ξ osc (φ ) 0,k ˆ 0,k ⎝ P ⎠ E(L ) ≤ 4E(L )E 2 e e . 1,1 3,1 P∈π (φ) −k If P is visited, then one of the four rectangles of size 2 (1, 3) in P surrounding P contains a short crossing, denoted by π ˜ (P) and we have ξφ (k) −k ξ inf φ −k ξφ (P) −ξ osc (φ ) 0,k ˆ 0,k 0,k ˆ 0,k P P e 1 ds ≥ L (π ˜ (P)) ≥ 2 e ≥ 2 e e , ˆ k π ∩P hence −k ξφ (P) ξ osc (φ ) 2ξ osc (φ ) ξφ 0,k ˆ 0,k ˆ 0,k 0,k P P 2 e e ≤ e e 1 ds. π ∩P k k P∈π (φ) P∈π (φ) k k Taking the supremum of the oscillation over all blocks, −k 2ξ 2 ∇φ (k) 2ξ osc (φ ) ξφ 0,k 0,k 0,k 2 ˆ [0,1] e e 1 ds ≤ 9e L . π ∩P 1,1 P∈π (φ) k TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 391 Altogether, by Cauchy–Schwarz, −k (n+k) (n) (k) 4ξ 2 ∇φ 2 1/2 0,k 1/2 [0,1] E(L ) ≤ 36E(L )E((L ) ) E(e ) . 1,1 3,1 1,1 When ξ satisﬁes Condition (T), by using the uniform bounds for quantile ratios together C k with the upper tail estimates (4.8) and the gradient estimate (2.10)weget λ ≤ e λ λ . n+k n k Step 2: Weak supermultiplicativity. We argue here that −C k (5.30) λ ≥ e λ λ . n+k n k Using a slightly easier argument than (5.15) (since we just have the ﬁeld φ here), we have −ξ max osc (φ ) 1 0,k (n+k) ˆ P∈P P (k,k+n) S ξφ (P) 0,k L ≥ e min L (R (P)) e , 1,1 i P∈P ,1≤i≤4 P∈π n+k where π denotes the k-coarse grained approximation of π , the left–right geodesic of n+k n+k 2 S [0, 1] for the ﬁeld φ ,and wherewerecallthat (R (P)) denote the four rectan- 0,n+k 1≤i≤4 −k gles of size 2 (1, 3) surrounding P. Furthermore, by using a similar argument to (5.3), we have (k) ξφ (P) −ξ max osc (φ ) k 0,k P∈P P 0,k e ≥ e 2 L . 1,1 P∈π n+k Altogether, we get the following weak supermultiplicativity, (n+k) (k) k (k,k+n) S −2ξ max osc (φ ) P∈P P 0,k (5.31)L ≥ L min 2 L (R (P)) e 1,1 1,1 P∈P ,1≤i≤4 When ξ satisﬁes Condition (T), by scaling and the tail estimates (4.7), k (k,k+n) S −C k −ck P(min 1 2 L (R (P)) ≥ λ e ) ≥ 1 − e . Furthermore, using the gradient P∈P ,1≤i≤4 n −k −ck estimates (2.9), we get P(2 ∇φ ≥ C k) ≥ 1 − e for C large enough. There- 0,k 2 [0,1] √ √ (n) −C k −C k fore, with probability ≥ 1/2, L ≤ e λ λ hence the bound λ ≥ e λ λ . n k n+k k n 1,1 5.3. Tightness of the log of the diameter. 2 −1 ξφ 0,n Proposition 27. — If γ ∈ (0, 2) and ξ = γ/d then log Diam [0, 1] ,λ e ds n≥0 is tight. Proof. — Step 1: Chaining. By a standard chaining argument, (see (6.1) in [18]for more details), we have 2 ξφ 2 (n) −n ξ sup φ 0,n 2 0,n [0,1] (5.32)Diam [0, 1] , e ds ≤ C max L (P) + C × 2 e , P∈C k=0 k −k where C is a collection of no more than C4 long rectangles of side length 2 (3, 1). k 392 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET ξ sup φ −n 2 0,n [0,1] Using the bound for the maximum (2.4), when ξ< 2, we have E(2 e ) ≤ −n 2ξ n C n 2 2 e . (n) Fix 0 ≤ k ≤ n and P ∈ C . We can bound L (P) by taking a left–right geodesic π k k,n for φ . Therefore, k,n (n) (n) ξ max φ (k,n) 2 0,k [0,1] L (P) ≤ L (π ) ≤ e L (P), k,n and consequently, (n) ξ max φ (k,n) 2 0,k [0,1] (5.33)max L (P) ≤ e max L (P). P∈C P∈C k k Using independence, the maximum bound (2.4), scaling of the ﬁeld φ and the tail estimates (4.8), we get +ε ξ max φ 2 (k,n) −k 2ξ k C k Ck 2 0,k [0,1] (5.34) E e max L (P) ≤ 2 2 e λ e n−k P∈C forsomeﬁxedsmall ε> 0 (again, the term k could in fact be log k). Taking the expec- tation in (5.32), using (5.33)and (5.34), we obtain the following bound for the expected value of the diameter, +ε 2 ξφ −k 2ξ k Ck 0,n (5.35) E(Diam([0, 1] , e ds)) ≤ C 2 2 λ e . n−k k=0 C k k(1−ξ Q) C k Step 2: Right tail. By Proposition 26, λ ≤ λ ≤ λ 2 e . Together with n−k n n (5.35), this implies that +ε 2 ξφ −k 2ξ k Ck 0,n E(Diam([0, 1] , e ds)) ≤ C 2 2 λ e n−k k=0 +ε −kξ(Q−2) Ck ≤ λ C 2 e . k=0 2 −1 ξφ s −s 0,n Since Q > 2, Markov’s inequality gives P Diam([0, 1] ,λ e ds) ≥ e ≤ Ce . Step 3: Left tail. Finally, since the diameter of the square [0, 1] is larger than the 2 −1 ξφ −s 0,n left–right distance, by our tail estimates (4.7), we get P Diam([0, 1] ,λ e ds) ≤ e ≤ (n) −s −cs P L ≤ λ e ≤ Ce . 1,1 TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 393 5.4. Tightness of the metrics. −1 ξφ 0,n Proposition 28. — If γ ∈ (0, 2) and ξ = γ/d then the sequence of metrics λ e ds n≥0 is tight. Moreover, if we deﬁne |x − x | d (x, x ) 0,n n n C := sup and C := sup α β 2 d (x, x ) 2 |x − x | 0,n x,x ∈[0,1] x,x ∈[0,1] n n then, for α> ξ(Q + 2) and β< ξ(Q − 2), the sequence (C , C ) is tight. n≥0 α β −1 ξφ 0,n Henceforth, we use the notation d for the renormalized metric λ e ds re- 0,n stricted to [0, 1] . Proof. — Theproof hastwo parts. In theﬁrstpartweshowthe tightness of the 2 2 + metrics in the space of continuous function from [0, 1] ×[0, 1] → R and in the second part we show that subsequential limits are metrics. A byproduct result of the argument is explicit bi-Hölder bounds. Part 1. Upper bound on the modulus of continuity. We suppose γ ∈ (0, 2).We start by proving that for every 0 <β <ξ(Q − 2),if ε> 0, there exists a large C > 0so that for every n ≥ 0 2 β (5.36) P ∃x, x ∈[0, 1] : d (x, x ) ≥ C |x − x | ≤ ε, 0,n ε β 2 i.e. d is tight, where the C -norm is deﬁned for f :[0, 1] × 0,n β 2 2 C ([0,1] ×[0,1] ) n≥0 [0, 1] → R as β 2 2 C ([0,1] ×[0,1] ) |f (x, y) − f (x , y )| := f + sup . 2 2 [0,1] ×[0,1] |(x, y) − (x , y )| 2 2 (x,y)=(x ,y )∈[0,1] ×[0,1] −n s By a union bound it sufﬁces to estimate P(∃x, x :|x − x | < 2 , d (x, x ) ≥ e |x − 0,n x | ) and −k −k+1 s β P ∃x, x : 2 ≤|x − x |≤ 2 , d (x, x ) ≥ e |x − x | . 0,n k=0 −k −k+1 s Step 1: We start with the term P(∃x, x : 2 ≤|x − x |≤ 2 , d (x, x ) ≥ e |x − 0,n x | ). We use the chaining argument (5.32)atscale k which gives ξ sup φ 0,n −1 (n) −1 −n 2 [0,1] sup d (x, x ) ≤ Cλ max L (P) + Cλ × 2 e . 0,n n n P∈C −k −k+1 i 2 ≤|x−x |≤2 i=k 394 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Taking the expected value and using the same bounds as those obtained in the proof of Proposition 27,weget 1 1 +ε +ε 2 2 −iξ(Q−2) Ci −kξ(Q−2) Ck E sup d (x, x ) ≤ 2 e ≤ C2 e . 0,n −k −k+1 2 ≤|x−x |≤2 i=k Therefore, using Markov’s inequality we get the bound −k −k+1 s β P ∃x, x : 2 ≤|x − x |≤ 2 , d (x, x ) ≥ e |x − x | 0,n k=0 n n s −kβ −s kβ −kξ(Q−2) ≤ P sup d (x, x ) ≥ e 2 ≤ e 2 2 . 0,n −k −k+1 2 ≤|x−x |≤2 k=0 k=0 The series is convergent since ξ(Q − 2) − β> 0. −n s β Step 2: We bound from above P(∃x, x |x − x | < 2 , d (x, x ) ≥ e |x − x | ) using 0,n a bound on the supremum of the ﬁeld. Indeed, for such x and x ,notethat ξ sup φ s β −1 2 0,n [0,1] e |x − x | ≤ d (x, x ) ≤ λ e |x − x | 0,n Writing β = ξ(Q − 2) − εξ for some ε> 0, it follows that 1 − β = (1 − ξ Q + 2ξ) + εξ > 0 since the LFPP exponent 1 − ξ Q ≥−2ξ by a simple uniform bound. Therefore, β−1 n(1−β) −1 n(1−β) n(2ξ +εξ +o(1)) |x − x | ≥ 2 and λ 2 = 2 . Altogether, this probability is bounded −1 from above by P(sup φ ≥ n log 4 + εn log 2 + o(n) + ξ s) and using (2.3)gives a 2 0,n [0,1] uniform tail estimate. Therefore, we obtain the tightness of d as a random element of C([0, 1] × 0,n n≥0 2 + [0, 1] , R ) and every subsequential limit is (by Skorohod’s representation theorem) a pseudo-metric. Part 2. Lower bound on the modulus of continuity. We prove that if α> ξ(Q + 2) and ε> 0 then there exists a small constant c > 0such that for every n ≥ 0, 2 α (5.37) P ∃x, x ∈[0, 1] : d (x, x ) ≤ c |x − x | ≤ ε. 0,n ε Similarly as before, by union bound it is enough to estimate the term 2 −n −ξ s α (5.38) P(∃x, x ∈[0, 1] :|x − x | < 2 , d (x, x ) ≤ e |x − x | ) 0,n and the term ⎛ ⎞ ⎜ ⎟ −k −k+1 −ξ s α (5.39) P ∃x, x : 2 ≤|x − x |≤ 2 , d (x, x ) ≤ e |x − x | . ⎝ 0,n ⎠ k=0 :=E k,n,s TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 395 2 −k Step 1: We give an upper bound for (5.39). Fix x, x ∈[0, 1] such that 2 ≤|x − −k+1 x |≤ 2 . Note that any path from x to x crosses one of the rectangles in the collection S 1 {R (P) : P ∈ P , 1 ≤ i ≤ 4}. Hence, under the event E , there exists x, x such that k,n,s i k+2 −kα −1 −k ξ inf φ k (k,n) S 2 0,k [0,1] (5.40)2 ≥ d (x, x ) ≥ λ 2 e min 2 L (R (P)) . 0,n n i P∈P ,1≤i≤4 k+2 Since α = ξ(Q + 2) + ξδ for a small δ> 0, by using Proposition 26 we get √ √ √ −kα k −kα C k −k(α−ξ Q) C k −k(2+δ)ξ −ξδk C k (5.41)2 λ 2 ≤ 2 λ λ e ≤ 2 λ e = 2 (λ 2 e ) n k n−k n−k n−k Now, using (5.40), (5.41) and scaling, we get ξ inf φ k (k,n) S −kα k −ξ s 2 0,k [0,1] P E ≤ P e min 2 L (R (P)) ≤ 2 λ 2 e k,n,s n P∈P ,1≤i≤4 k+2 ≤ P sup |φ |≥ k log 4 + kδ log 2 + s/2 0,k [0,1] (n−k) S −kδξ C k −ξ s/2 + P min L (R (P)) ≤ λ 2 e e n−k P∈P ,1≤i≤4 k+2 −ck −cs ≤ Ce e , where we used in the last inequality the supremum bounds (2.3) and the left tail estimate (4.7). Step 2: Finally, we control (5.38). We write −n −ξ s α P(∃x, x :|x − x | < 2 , d (x, x ) ≤ e |x − x | ) 0,n d (x, x ) 0,n −ξ s ≤ P inf ≤ e −n |x−x |≤2 |x − x | −1 ξ inf φ 1−α −ξ s 2 0,n [0,1] ≤ P λ e inf |x − x | ≤ e . −n |x−x |≤2 We recall that α> ξ Q + 2ξ , and in particular α> 1: indeed, 1 − ξ Q ≤ 2ξ follows from 1−α −n(1−α) a comparison with the inﬁmum of the ﬁeld. In this case, inf −n |x − x | = 2 , |x−x |≤2 and by Proposition 26, √ √ −n(1−α) −1 −n(1−α) n(1−ξ Q) −C n n(α−ξ Q) −C n 2 λ ≥ 2 2 e = 2 e n 396 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET Therefore, since α − ξ Q = 2ξ + δξ for some δ> 0, we have for n large that ξ inf φ −1 1−α −ξ s 2 0,n [0,1] P λ e inf |x − x | ≤ e −n |x−x |≤2 ≤ P sup |φ |≥ n log 4 + n log 2 + s 0,n [0,1] Using (2.3) completes the proof. Acknowledgements We would like to thank the referees for many helpful comments which helped to improve the exposition of the manuscript. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. Appendix A A.1 Comparison with the GFF molliﬁed by the heat kernel Let h be a GFF with Dirichlet boundary condition on a domain D and U ⊂⊂ Dbe a subdomain of D. We recall that we denote by p the two-dimensional heat kernel at time t |x| 1 − 2t i.e. p (x) = e . The goal of this section is to obtain a uniform estimate to conclude on 2π t the tightness of the renormalized metric associated to p ∗ h assuming the one associated to φ . In particular, the second assertion of Theorem 1 is a corollary of the following proposition. Proposition 29. — There exist constants C, c > 0 such that for all t ∈ (0, 1/2), there is a (d) coupling of h and ϕ = φ such that for all x ≥ 0, we have −cx P ϕ − p ∗ h ≥ x ≤ Ce . U TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 397 Molliﬁcation of the GFF by the heat kernel. — The covariance of the Gaussian ﬁeld p ∗ h is given for x, x ∈ Uby t t t t E p ∗ h(x) p ∗ h(x ) = p (x − y)G (y, y )p (y − x )dydy , 2 2 2 2 D D where G is the Green function associated to the Laplacian operator on D. For an open set A, we denote by p (x, y) the transition probability density of a Brownian motion killed upon exiting A. White noise representation. — Take a space–time white noise W and deﬁne the ﬁeld η on Uby (A.1) η (x) := p ∗ p s (x, y)W(dy, ds) 0 D D D t t where p ∗ p s (x, y) := p (x − y )p s (y , y)dy , 2 2 2 2 (d) so that (η (x)) = (p t ∗ h(x)) . Indeed, by Fubini, we have t x∈U x∈U E(η (x)η (x )) t t D D t t = p ∗ p (x, y) p ∗ p (x , y)dyds s s 2 2 2 2 0 D D D t t = p (x − y )p s (y , y) p (x − y ) p s (y , y)dydy dy ds 2 2 2 2 0 D D D D D t t = p (x − y ) p s (y , y)p s (y, y )dyds p (x − y )dy dy 2 2 2 2 D D 0 D t t = p (x − y )G (y , y )p (y − x )dy dy . 2 2 D D 1 (d) Coupling. — Note that for t ∈ (0, 1/2)φ (x) = p (x − y)W(dy, ds) = ϕ (x),where 2 t t R 2 we set 1−t t+s ϕ (x) := p (x − y)W(dy, ds). 0 R 1 2 Furthermore, we can decompose ϕ (x) = ϕ (x) + ϕ (x),where t t 1−t t+s (A.2) ϕ (x) := p (x − y)W(dy, ds); 0 D 1−t t+s (A.3) ϕ (x) := p (x − y)W(dy, ds). 0 D 398 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET 1 2 Recalling the deﬁnition of η in (A.1), we introduce η and η so that t t 1−t ∞ D D t t (A.4) η (x) = p ∗ p s (x, y)W(dy, ds) + p ∗ p s (x, y)W(dy, ds) 2 2 2 2 0 D 1−t D 1 2 =: η (x) + η (x). t t Therefore, under this coupling (viz. using the same white noise W), we have 1−t 1 1 D (A.5) ϕ (x) − η (x) = p t+s (x − y) − p ∗ p (x, y) W(dy, ds). t t 2 2 0 D Comparison between kernels. — We will consider x, y ∈ U, subdomain of D. Set d := d(U, D )> 0. D D D t t t p ∗ p s (x, y) := p (x − y )p s (y , y)dy = p (x − y )p (y − y)q s (y , y)dy , 2 2 2 2 2 2 D D where q (x, x ) is the probability that a Brownian bridge between x and x with lifetime t stays in D. Therefore, using Chapman–Kolmogorov, t t+s t s p ∗ p s (x, y) − p (x, y) =− p (x − y )p (y − y)dy 2 2 2 t s + p (x − y )p (y − y)(q s (y , y) − 1)dy . 2 2 Note that the ﬁrst term can be bounded by using that |y − y |≥ d for y ∈ Uand y ∈ D . For the second term, we can split the integral over D in two parts: one over the ε- neighborhood of ∂ D (within D), denoted by (∂ D) , and one over its complement. To give ε ε an upper bound on the ﬁrst, we use that for y ∈ Uand y ∈ (∂ D) , |y − y |≥ d(U,(∂ D) ). Finally, we bound the second part by using a uniform estimate on the probability that a Brownian bridge between a point in U and a point D \ (∂ D) exits D in time less than s/2. (Note that 1 − q s (y, y ) is the probability that a Brownian bridge between y and y with time length s/2 exits D.) Therefore, we get that uniformly in x, y ∈ Uand t, D − (A.6) |p ∗ p (x, y) − p t+s (x, y)|≤ Ce . 2 2 Comparison between ϕ and p ∗ h. — By the triangle inequality, 1 1 2 2 (A.7) ϕ − p ∗ h ≤ ϕ − η + ϕ + η . t t t t U U U We look for a uniform right tail estimate (in t) of each term in the right-hand side of (A.7). In order to do so, we will use the Kolmogorov continuity criterion. Therefore, we derive below some pointwise and difference estimates. TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 399 First term. We derive ﬁrst a pointwise estimate. For x ∈ U, using the kernel comparison (A.6), there exists some C > 0 such that, uniformly in t, 1−t 1 2 D t t+s Var η (x) − ϕ (x) = p ∗ p s (x, y) − p (x, y) dyds t t 2 2 0 D 1−t ≤ C e ds ≤ C . 1 1 We now give a difference estimate: introducing (x) := ϕ (x) − η (x),for x, x ∈ U, t t 1−t E (x) − x = p t+s (x − y) − p t ∗ p (x, y) t t 2 2 0 D t+s t − p x − y − p ∗ p s x , y dyds, 2 2 which is uniformly bounded in t ∈ (0, 1/2) by a quantity of size O(|x − x |). (By splitting the integral at |x − x |,one canuse (A.6) for the small values of s and gradient estimates for both kernels for larger values of s.) Second term. We recall here that ϕ (x) is deﬁned for x ∈ Uby 1−t 1 (d) t+s ϕ (x) = p (x − y)W(dy, ds) = p (x − y)W(dy, ds). 2 2 c c 0 D t D We have, for x, x ∈ U, with d := d(U, D ), 2 2 E ϕ (x) − ϕ (x ) t t s s ≤ p (x − y) − p (x − y) dyds 2 2 t D s s ≤ p (x − y) − p (x − y) dyds 2 2 |x−x | R |x−x | s s + p (x − y) − p (x − y) dyds 2 2 0 D 1 |x−x | ≤ 2 p (0) − p (x − x ) ds + 4 p (d)ds ≤ C|x − x |, s s |x−x | 0 −z whereweuse 1 − e ≤ z in the last inequality. Similarly, we can prove that there exists C > 0 independent of t such that E(φ (x) ) ≤ C. 2 2 Third term. We recall here that η (x) is deﬁned for x ∈ Uby η (x) = p t ∗ t t 1−t D p s (x, y)W(dy, ds). Similarly, there exists C > 0such that for t ∈ (0, 1/2), x, x ∈ U, we 2 400 JIAN DING, JULIEN DUBÉDAT, ALEXANDER DUNLAP, HUGO FALCONET have 2 2 D D t t E η (x) − η (x ) ≤ p ∗ p s (x, y) − p ∗ p s (x , y) dyds t t 2 2 2 2 1/2 D ≤ C|x − x |. Furthermore, the pointwise variance is uniformly bounded. Result. Altogether, coming back to (A.7) and combining Kolmogorov continuity criterion with Fernique’s theorem (see Section 1.3 in [22]), we get the following tail esti- mate on the above coupling: there exist C, c > 0such that for all t ∈ (0, 1/2), x ≥ 0, we have −cx P ϕ − p ∗ h ≥ x ≤ Ce . A.2 Approximations for δ ∈ (0, 1) −n We explain here how results obtained along the sequence {2 : n ≥ 0} can be extended −(n+r) to δ ∈ (0, 1).For each δ ∈ (0, 1),let n ≥ 0and r ∈[0, 1] such that δ = 2 .Thenby decoupling the ﬁeld φ , using a uniform estimate for r ∈[0, 1] and a scaling argument, 0,r −n we generalize our previous results obtained along the sequence 2 to δ ∈ (0, 1). Decoupling low frequency noise. — Note that there exists C > 0such that for n ≥ 0and r ∈[0, 1] we have −C C (A.8) e λ ≤ λ ≤ λ e . n n+r n (r,n+r) (n+r) (r,n+r) −ξ inf φ ξ sup φ 2 0,r 2 0,r [0,1] [0,1] Indeed, note that a.s. e L ≤ L ≤ e L . Furthermore, with 1,1 1,1 1,1 (d) (r,n+r) (n) −r high probability sup |φ |≤ C ≤ C. Then, note that L = 2 L and a.s. 2 r r 0,r r [0,1] 1,1 2 ,2 (n) (n) (n) L ≤ L ≤ L . By the tightness result, there exists a constant C > 0 such that uni- r r 1,2 2 ,2 2,1 (n) (n) −C C formly in n, with high probability, L ≥ e λ and L ≤ e λ , therefore, with high n n 1,2 2,1 (r,n+r) −C C probability, e λ ≤ L ≤ e λ ,hence (A.8). n n 1,1 Weak multiplicativity. — In this paragraph, we will use the notation λ from the introduc- −n tion. We recall that writing λ instead of λ was an abuse of notation. Now we prove n 2 that there exists C > 0such that for δ, δ ∈ (0, 1) we have √ √ −1 −C | log δ∨δ | C | log δ∨δ | (A.9)C e λ λ ≤ λ ≤ Ce λ λ . δ δ δδ δ δ Similarly as (A.8), there exists C > 0such that for r, r ∈[0, 1], n, n ≥ 0, −C C (A.10) e λ ≤ λ ≤ λ e . −n−n −n−r−n −r −n−n 2 2 2 TIGHTNESS OF LIOUVILLE FIRST PASSAGE PERCOLATION FOR γ ∈ (0, 2) 401 −(n+r) −(n +r ) For δ, δ ∈ (0, 1),let n, n ≥ 0and r, r ∈[0, 1] such that δ = 2 , δ = 2 .Note that n =[− log δ]. Using the weak multiplicativity for powers of 2, we have √ √ −C n∧n C n∧n −n −n (A.11) e λ λ −n ≤ λ −n−n ≤ λ λ −n e . 2 2 2 2 2 Without loss of generality, we consider just the upper bound in (A.9). The lower bound follows along the same lines. By using ﬁrst (A.10)and then (A.11)weget C C n∧n C λ = λ ≤ λ e ≤ λ −n λ e e . −n−r−n −r −n−n −n δδ 2 2 2 2 Now, the result follows by using (A.8): √ √ C n∧n C n+r∧n +r 2C C log |δ∨δ | 2C −n −n−r λ λ e ≤ λ λ e e = λ λ e e . −n −n −r 2 2 δ δ 2 2 Tail estimates and tightness of metrics. — Using the same argument as in the two previous −n paragraphs and the tail estimates obtained along the sequence {2 : n ≥ 1},wehavethe following tail estimates for crossing lengths of the rectangles [0, a]×[0, b]: there exists c, C > 0 (depending only on a, b and γ )suchthatfor s > 2, uniformly in δ ∈ (0, 1),we have −1 (δ) s −c log s (A.12) P λ L ≥ e ≤ Ce ; δ a,b −1 (δ) −s −cs (A.13) P λ L ≤ e ≤ Ce . δ a,b −1 ξφ 2 Furthermore, the sequence of metrics (λ e ds) on [0, 1] is tight. δ∈(0,1) REFERENCES 1. M. ANG, Comparison of discrete and continuum Liouville ﬁrst passage percolation, Electron. Commun. Probab., 24 (2019), 2. J. ARU,E.POWELL and A. 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Department of Mathematics, Stanford University, Stanford, CA 94305, USA Current address: Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA H. F. Department of Mathematics, Columbia University, New York, NY 10027, USA Manuscrit reçu le 20 novembre 2019 Version révisée le 23 octobre 2020 Manuscrit accepté le 30 octobre 2020 publié en ligne le 4 novembre 2020.

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