Almost invariance of distributions for random walks on groups

TL;DR

This paper investigates the near-invariance of transition probabilities in random walks on groups, establishing conditions, bounds, and examples of almost invariant distributions, with implications for group properties and random walk behavior.

Contribution

It proves that simple random walks on wreath products have almost invariant distributions and introduces the concept of the radius of almost invariance, providing bounds and examples.

Findings

Wreath product random walks exhibit almost invariant distributions.

The radius of almost invariance can be asymptotically smaller or larger than certain functions.

Existence of groups with unbounded but controlled radius of invariance.

Abstract

We study the neighborhoods of a typical point $Z_n$ visited at $n$-th step of a random walk, determined by the condition that the transition probabilities stay close to $\mu^{*n}(Z_n)$. If such neighborhood contains a ball of radius $C \sqrt{n}$, we say that the random walk has almost invariant transition probabilities. We prove that simple random walks on wreath products of $\mathbb{Z}$ with finite groups have almost invariant distributions. A weaker version of almost invariance implies a necessary condition of Ozawa's criterion for the property $H_{\rm FD}$. We define and study the radius of almost invariance, we estimate this radius for random walks on iterated wreath products and show this radius can be asymptotically strictly smaller than $n/L(n)$, where $L(n)$ denotes the drift function of the random walk. We show that the radius of individual almost invariance of a simple random walk on the wreath product of $\mathbb{Z}^2$ with a finite group is asymptotically strictly larger than $n/L(n)$. Finally, we show the existence of groups such that the radius of almost invariance is smaller than a given function, but remains unbounded. We also discuss possible limiting distribution of ratios of transition probabilities on non almost invariant scales.