Continuous phase transitions on Galton-Watson trees

DOI10.1017/S0963548321000237zbMATH Open1511.60125arXiv2007.13864MaRDI QIDQ6345958

Author name not available (Why is that?)

Publication date: 27 July 2020

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

m a t h c a l T_{1}

be the event that a Galton-Watson tree is infinite, and let

m a t h c a l T_{2}

be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties:

m a t h c a l T_{1}

holds if and only if

m a t h c a l T_{1}

holds for at least one of the trees initiated by children of the root, and

m a t h c a l T_{2}

holds if and only if

m a t h c a l T_{2}

holds for at least two of these trees. The probability of

m a t h c a l 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

m a t h c a l 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.

Mathematics Subject Classification ID

No records found.

This page was built for publication: Continuous phase transitions on Galton-Watson trees

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6345958)