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 T1 be the event that a Galton-Watson tree is infinite, and let T2 be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: T1 holds if and only if T1 holds for at least one of the trees initiated by children of the root, and T2 holds if and only if T2 holds for at least two of these trees. The probability of T1 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 T2 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.