Monotonicity of the speed for biased random walk on Galton-Watson tree

TL;DR

This paper extends the understanding of how the speed of biased random walks on Galton-Watson trees decreases with bias parameter, providing a broader range of parameters for which the speed is strictly decreasing.

Contribution

It establishes a new, larger interval for the bias parameter where the speed of the walk is strictly decreasing, generalizing previous results.

Findings

Speed is strictly decreasing for            when           .

The result applies to Galton-Watson trees without leaves with minimal degree m_1  2 or greater.

The proof relies on techniques from A"ide9kon's work on biased random walks.

Abstract

Ben Arous, Fribergh and Sidoravicius \cite{GAV2014} proved that speed of biased random walk $RW_\lambda$ on a Galton-Watson tree without leaves is strictly decreasing for $\lambda\leq \frac{m_1}{1160},$ where $m_1$ is minimal degree of the Galton-Watson tree. And A\"{\i}d\'{e}kon \cite{EA2013} improved this result to $\lambda\leq \frac{1}{2}.$ In this paper, we prove that for the $RW_{\lambda}$ on a Galton-Watson tree without leaves, its speed is strictly decreasing for $\lambda\in \left[0,\frac{m_1}{1+\sqrt{1-\frac{1}{m_1}}}\right]$ when $m_1\geq 2;$ and we owe the proof to A\"{\i}d\'{e}kon \cite{EA2013}.