Quenched scaling limit of critical percolation clusters on Galton-Watson trees

TL;DR

This paper proves that quenched critical percolation clusters on Galton-Watson trees with stable offspring tails converge to stable trees, and the random walk on these clusters converges to Brownian motion on the stable tree.

Contribution

It establishes the GHP scaling limit of quenched critical percolation clusters on Galton-Watson trees with stable offspring tails as the corresponding stable tree, extending previous annealed results.

Findings

Quenched critical percolation clusters scale to stable trees.

Random walk on clusters converges to Brownian motion on the stable tree.

Provides quenched asymptotics for cluster size tail distribution.

Abstract

We consider quenched critical percolation on a supercritical Galton--Watson tree with either finite variance or $\alpha$-stable offspring tails for some $\alpha \in (1,2)$. We show that the GHP scaling limit of a quenched critical percolation cluster on this tree is the corresponding $\alpha$-stable tree, as is the case in the annealed setting. As a corollary we obtain that a simple random walk on the cluster also rescales to Brownian motion on the stable tree. Along the way, we also obtain quenched asymptotics for the tail of the cluster size, which completes earlier results obtained in Michelen (2019) and Archer-Vogel (2024).