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Continuous phase transitions on Galton–Watson trees

Part of: Equilibrium statistical mechanics Special processes Markov processes

Published online by Cambridge University Press:  06 July 2021

Tobias Johnson
Tobias Johnson*
Affiliation:
Department of Mathematics, College of Staten Island, Staten Island, NY 10314, USA
*
Email: tobias.johnson@csi.cuny.edu
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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.

Keywords

Galton–Watson treephase transitionfirst-order

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B26: Phase transitions (general)
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 2 , March 2022 , pp. 198 - 228
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

The author received support from NSF grant DMS-1811952 and PSC-CUNY Award #62628-00 50.

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