Rate of Escape of Random Walks on Regular Languages and Free Products by Amalgamation of Finite Groups

TL;DR

This paper analyzes the escape rate of specific random walks on words over finite alphabets and free products of finite groups, providing formulas for their escape rates using various methods.

Contribution

It introduces new formulas for the escape rate of these random walks, connecting their behavior to the structure of free products by amalgamation of finite groups.

Findings

Derived multiple formulas for the escape rate.

Connected escape rates to group structures.

Applied different techniques for computation.

Abstract

We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques.