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Continuous phase transitions on Galton–Watson trees

Continuous phase transitions on Galton–Watson trees Abstract Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton–Watson trees. For example, let $\mathcal{T}_1$ be the event that a Galton–Watson tree is infinite and let $\mathcal{T}_2$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $\mathcal{T}_1$ holds if and only if $\mathcal{T}_1$ holds for at least one of the trees initiated by children of the root, and $\mathcal{T}_2$ holds if and only if $\mathcal{T}_2$ holds for at least two of these trees. The probability of $\mathcal{T}_1$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $\mathcal{T}_2$ has a first-order phase transition, jumping discontinuously to a non-zero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterise the event undergoing the phase transition. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Combinatorics Probability Computing Cambridge University Press

Continuous phase transitions on Galton–Watson trees

Combinatorics Probability Computing , Volume 31 (2): 31 – Mar 1, 2022

Continuous phase transitions on Galton–Watson trees

s1.sIntroductionsUnderstanding phase transitions is a central task in discrete probability and statistical physics. One of the most basic questions about a phase transition is whether it is continuous or first-order. That is, when a quantity undergoes a phase transition, does it vary continuously as a parameter is varied, or does it take a discontinuous jump at criticality? This question is often difficult. For example, the phase transition for the probability that the origin belongs to an infinite component in bond percolation on the lattice is thought to be continuous, but it remains unproven in dimensions s$3,\ldots,10$s[10, 13].sThis paper investigates phase transitions on Galton–Watson trees for events satisfying certain recursive properties. This setting is inspired by two examples. Let s$\mathcal{T}_1$sbe the set of infinite rooted trees and let s$\mathcal{T}_2$sbe the set of trees containing an infinite binary tree starting from the root. Let s$T_\lambda$sbe a Galton–Watson tree with child distribution s$\textrm{Poi}(\lambda)$s. The event s$\{T_\lambda\in\mathcal{T}_1\}$shas probability 0 for s$\lambda<1$s. It undergoes a continuous phase transition at s$\lambda=1$s, with its probability rising from 0 as s$\lambda$sincreases above 1. On the other hand, the event s$\{T_\lambda\in\mathcal{T}_2\}$shas probability 0 for s$\lambda<\lambda_{\textrm{crit}}\approx 3.35$s. Its probability jumps to approximately 0.535 at s$\lambda=\lambda_{\textrm{crit}}$sand increases continuously after that. See [14, Example 5.5] for a detailed treatment of this example; see [18] for this example in the context of random graphs; and see [6] for an earlier analysis of s$\mathcal{T}_2$sand proof that the phase transition is discontinuous for a different family of child distributions.sThe sets s$\mathcal{T}_1$sand s$\mathcal{T}_2$sboth satisfy recursive properties. A tree is in s$\mathcal{T}_1$sif and only if the root has at least one child that initiates a tree in...
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References (24)

Publisher
Cambridge University Press
Copyright
© The Author(s), 2021. Published by Cambridge University Press
ISSN
1469-2163
eISSN
0963-5483
DOI
10.1017/S0963548321000237
Publisher site
See Article on Publisher Site

Abstract

Abstract Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton–Watson trees. For example, let $\mathcal{T}_1$ be the event that a Galton–Watson tree is infinite and let $\mathcal{T}_2$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $\mathcal{T}_1$ holds if and only if $\mathcal{T}_1$ holds for at least one of the trees initiated by children of the root, and $\mathcal{T}_2$ holds if and only if $\mathcal{T}_2$ holds for at least two of these trees. The probability of $\mathcal{T}_1$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $\mathcal{T}_2$ has a first-order phase transition, jumping discontinuously to a non-zero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterise the event undergoing the phase transition.

Journal

Combinatorics Probability ComputingCambridge University Press

Published: Mar 1, 2022

Keywords: Galton–Watson tree; phase transition; first-order; 60J80; 60K35; 82B26

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