Markovian linearization of random walks on groups

TL;DR

This paper introduces a new linearization technique for analyzing finitely supported random walks on groups by transforming them into simpler nearest-neighbor colored walks, enabling explicit calculations of drift and entropy.

Contribution

It develops a novel linearization method tailored for random walks on groups, extending known results to colored walks and providing explicit formulas for key quantities.

Findings

Extended drift and entropy formulas to colored random walks

Simplified analysis of random walks on free groups and free products

Introduced a linearization approach for non-commutative polynomial evaluation

Abstract

In operator algebra, the linearization trick is a technique that reduces the study of a non-commutative polynomial evaluated at elements of an algebra A to the study of a polynomial of degree one, evaluated on the enlarged algebra A x M r (C), for some integer r. We introduce a new instance of the linearization trick which is tailored to study a finitely supported random walk on a group G by studying instead a nearest-neighbor colored random walk on G x {1,. .. , r}, which is much simpler to analyze. As an application we extend well-known results for nearest-neighbor walks on free groups and free products of finite groups to colored random walks, thus showing how one can obtain explicit formulas for the drift and entropy of a finitely supported random walk.