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 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.

arXiv commentminor small corrections; 34 pages; to appear in Combinatorics, Probability and Computing

References in corpus (6)

• Explosive transitions in complex networks' structure and dynamics: percolation and synchronization
• The k-core and branching processes
• A short proof of the Harris-Kesten Theorem
• Fundamental ingredients for the emergence of discontinuous phase transitions in the majority vote model
• Random tree recursions: which fixed points correspond to tangible sets of trees?
• Pemantle's min-plus binary tree