Critical Percolation and the Incipient Infinite Cluster on Galton-Watson Trees

TL;DR

This paper investigates critical percolation on Galton-Watson trees, establishing asymptotic probabilities, distributional limits, and the construction of the incipient infinite cluster, with results depending solely on the offspring distribution.

Contribution

It provides quenched analogues of classical branching process theorems for critical percolation on Galton-Watson trees, including new limit laws and the construction of the IIC.

Findings

Percolation reaches depth n with probability asymptotic to a constant times n^{-1}.

Number of vertices at depth n conditioned on reaching n converges to an exponential distribution.

Constructs the incipient infinite cluster and establishes a limit law for vertices at depth n.

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.