The speed of random walk on Galton-Watson trees with vanishing conductances

TL;DR

This paper investigates how the speed of a random walk on Galton-Watson trees varies with the distribution of conductances, especially as the conductance distribution approaches a non-elliptic limit, revealing insights into the walk's asymptotic behavior.

Contribution

It analyzes the regularity of the walk's speed as a function of conductance distribution, including cases with vanishing conductances, providing new understanding of the walk's asymptotic properties.

Findings

Speed satisfies a law of large numbers with a well-defined limit.

Regularity of the speed function depends on the conductance distribution.

Behavior when conductance distribution converges to a non-elliptic limit is characterized.

Abstract

In this paper we consider random walks on Galton-Watson trees with random conductances. On these trees, the distance of the walker to the root satisfies a law of large numbers with limit the effective velocity, or speed of the walk. We study the regularity of the speed as a function of the distribution of conductances, in particular when the distribution of conductances converges to a non-elliptic limit.