Continuous phase transitions on Galton-Watson trees

TL;DR

This paper investigates phase transitions in recursive events on Galton-Watson trees, distinguishing between continuous and first-order transitions, and characterizes the conditions under which these occur.

Contribution

It provides a framework to identify and analyze continuous versus first-order phase transitions for recursive events on Galton-Watson trees.

Findings

Probability of infinite Galton-Watson tree increases continuously above critical mean.

Probability of containing an infinite binary subtree jumps discontinuously at criticality.

Characterization of child distributions leading to continuous phase transitions.

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 nonzero 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 characterize the event undergoing the phase transition.