Quenched Survival of Bernoulli Percolation on Galton–Watson Trees

Marcus Michelen*, Robin Pemantle, Josh Rosenberg

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We explore the survival function for percolation on Galton–Watson trees. Letting g(T, p) represent the probability a tree T survives Bernoulli percolation with parameter p, we establish several results about the behavior of the random function g(T, · ) , where T is drawn from the Galton–Watson distribution. These include almost sure smoothness in the supercritical region; an expression for the kth -order Taylor expansion of g(T, · ) at criticality in terms of limits of martingales defined from T (this requires a moment condition depending on k); and a proof that the kth order derivative extends continuously to the critical value. Each of these results is shown to hold for almost every Galton–Watson tree.

Original languageEnglish
Pages (from-to)1323-1364
Number of pages42
JournalJournal of Statistical Physics
Volume181
Issue number4
DOIs
StatePublished - 1 Nov 2020
Externally publishedYes

Keywords

  • Branching process
  • Quenched survival
  • Random tree
  • Supercritical

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