The Speed of a Biased Walk on a Galton-Watson Tree without Leaves is Monotonic with Respect to Progeny Distributions for High Values of Bias

TL;DR

This paper demonstrates that for high bias values, the speed of a biased random walk on Galton-Watson trees without leaves increases monotonically with the progeny distribution, addressing a question in stochastic processes.

Contribution

It proves that the walk's speed is monotonic with respect to progeny distributions for high bias, partially answering an open question in the field.

Findings

Speed is greater on GW(P_1) than GW(P_2) for high bias when P_1 stochastically dominates P_2.

Monotonicity of speed holds above a certain bias threshold depending on progeny distributions.

Addresses an open problem posed by Ben Arous, Fribergh, and Sidoravicius.

Abstract

Consider biased random walks on two Galton-Watson trees without leaves having progeny distributions $P_1$ and $P_2$ (GW$(P_1)$ and GW$(P_2)$) where $P_1$ and $P_2$ are supported on positive integers and $P_1$ dominates $P_2$ stochastically. We prove that the speed of the walk on GW$(P_1)$ is bigger than the same on GW$(P_2)$ when the bias is larger than a threshold depending on $P_1$ and $P_2$. This partially answers a question raised by Ben Arous, Fribergh and Sidoravicius.