Central limit theorem for a random walk on Galton-Watson trees with random conductances

TL;DR

This paper establishes a central limit theorem for random walks on Galton-Watson trees with randomly assigned conductances, analyzing how small conductance edges affect the walk's variance and behavior.

Contribution

It introduces a CLT for such random walks with random conductances and examines the impact of small conductance edges on the limiting variance.

Findings

Variance remains nonvanishing as small conductance edges diminish, under supercritical conditions.

The slowdown effect of small conductance edges is limited, maintaining the walk's diffusive behavior.

A regeneration structure is used to derive uniform escape estimates regardless of conductance size.

Abstract

We show a central limit theorem for random walk on a Galton-Watson tree, when the edges of the tree are assigned randomly uniformly elliptic conductances. When a positive fraction of edges is assigned a small conductance $\varepsilon$, we study the behavior of the limiting variance as $\varepsilon\to 0$. Provided that the tree formed by larger conductances is supercritical, the variance is nonvanishing as $\varepsilon\to 0$, which implies that the slowdown induced by the $\varepsilon$-edges is not too strong. The proof utilizes a specific regeneration structure, which leads to escape estimates uniform in $\varepsilon$.