Effective Drift Estimates for Random Walks on Graph Products

TL;DR

This paper establishes uniform lower bounds on the drift for various random walks on graph products, including right-angled Artin groups, by extending existing arguments and combinatorial methods without moment conditions.

Contribution

It introduces a new approach to estimate drift bounds for random walks on graph products, broadening applicability to walks with minimal assumptions.

Findings

Uniform lower bounds on drift established for a large family of random walks.

Applicable to simple random walks on right-angled Artin groups with sparse graphs.

Method avoids moment conditions by alternating between specific measures.

Abstract

We find uniform lower bounds on the drift for a large family of random walks on graph products, of the form $ \mathbb{P} (|Z_{n}| \leq \kappa n) \leq e ^{-\kappa n} $ for $ \kappa > 0 $. This includes the simple random walk for a right-angled Artin group with a sparse defining graph. This is done by extending an argument of Gou\"{e}zel, along with the combinatorial notion of a piling introduced by Crisp, Godelle, and Wiest. We do not use any moment conditions, instead considering random walks which alternate between one measure uniformly distributed on vertex groups, and another measure over which we make almost no assumptions.