Random walks on Galton-Watson trees with random conductances

TL;DR

This paper studies random walks with random conductances on infinite Galton-Watson trees, proving a positive speed exists under finite mean conductance and providing a formula for it, with implications for comparison to simple random walks.

Contribution

It introduces a formula for the speed of random walks with random conductances on Galton-Watson trees and compares it to simple random walk speeds.

Findings

Existence of a positive deterministic speed under finite mean conductance.

Derived a formula for the speed based on effective conductances.

Speed is not larger than that of simple random walk when conductances have the same mean.

Abstract

We consider the random conductance model, where the underlying graph is an infinite supercritical Galton--Watson tree, the conductances are independent but their distribution may depend on the degree of the incident vertices. We prove that, if the mean conductance is finite, there is a deterministic, strictly positive speed $v$ such that $\lim_{n\to\infty} \frac{|X_n|}{n}= v$ a.s.\ (here, $|\cdot|$ stands for the distance from the root). We give a formula for $v$ in terms of the laws of certain effective conductances and show that, if the conductances share the same expected value, the speed is not larger than the speed of simple random walk on Galton--Watson trees. The proof relies on finding a reversible measure for the environment observed by the particle.