Random interlacements on Galton-Watson Trees

TL;DR

This paper investigates the critical percolation parameter for random interlacements on Galton-Watson trees, establishing its almost sure constancy and providing a characterization as a solution to a specific equation.

Contribution

It proves the critical parameter is almost surely constant and non-trivial, extending previous work and characterizing its value explicitly.

Findings

Critical parameter u* is almost surely constant.

u* is non-trivial for Galton-Watson trees.

Provides an explicit equation for u*.

Abstract

We study the critical parameter u^{*} of random interlacements percolation (introduced by A.S Sznitman in arXiv:0704.2560) on a Galton-Watson tree conditioned on the non-extinction event. Starting from the previous work of A. Teixeira in arXiv:0907.0316, we show that, for a given law of a Galton-Watson tree, the value of this parameter is a.s. constant and non-trivial. We also characterize this value as the solution of a certain equation.