Critical percolation and the incipient infinite cluster on Galton-Watson trees

Marcus Michelen

doi:10.1214/19-ecp216

Abstract

We consider critical percolation on Galton-Watson trees and prove quenched analogues of classical theorems of critical branching processes. We show that the probability critical percolation reaches depth $n$ is asymptotic to a tree-dependent constant times $n^{-1}$. Similarly, conditioned on critical percolation reaching depth $n$, the number of vertices at depth $n$ in the critical percolation cluster almost surely converges in distribution to an exponential random variable with mean depending only on the offspring distribution. The incipient infinite cluster (IIC) is constructed for a.e. Galton-Watson tree and we prove a limit law for the number of vertices in the IIC at depth $n$, again depending only on the offspring distribution. Provided the offspring distribution used to generate these Galton-Watson trees has all finite moments, each of these results holds almost-surely.

Highlights

We consider percolation on a locally finite rooted tree T : each edge is open with probability p ∈ (0, 1), independently of all others
We show that the probability critical percolation reaches depth n is asymptotic to a tree-dependent constant times n−1
Conditioned on critical percolation reaching depth n, the number of vertices at depth n in the critical percolation cluster almost surely converges in distribution to an exponential random variable with mean depending only on the offspring distribution

Summary

Introduction

We consider percolation on a locally finite rooted tree T : each edge is open with probability p ∈ (0, 1), independently of all others. These questions were answered in the study of critical branching processes These classical results apply to annealed critical percolation on Galton-Watson trees. For a Galton-Watson tree T , let Zn denote the number of vertices at distance of n from the root; the process Wn = Zn/(E[Z])n converges almost-surely to some random variable W. The limit laws of parts (b) and (c) of Theorem 1.3 do not depend at all on T itself but just on the distribution of Z This is in sharp contrast to the case of near-critical and supercritical percolation on Galton-Watson trees, in which the behavior is dependent on the tree itself [MPR18]. We suspect that less rigid conditions are sufficient, but this would require a different proof strategy than the method of moments, perhaps utilizing a stronger anti-concentration statement in the vein of Proposition 3.8

Set-up and notation

Moments

Survival probabilities

Incipient infinite cluster