Asymptotic Entropy of Random Walks on Regular Languages over a Finite Alphabet

TL;DR

This paper establishes the existence and formulas for the asymptotic entropy of random walks on regular languages over finite alphabets, showing its real-analytic dependence on certain probability measures, with applications to virtually free groups.

Contribution

It proves the existence of asymptotic entropy for these random walks and provides explicit formulas, also demonstrating the entropy's real-analytic variation with respect to probability measures.

Findings

Asymptotic entropy exists for random walks on regular languages.

Entropy varies real-analytically with probability measures.

Results apply to random walks on virtually free groups.

Abstract

We prove existence of asymptotic entropy of random walks on regular languages over a finite alphabet and we give formulas for it. Furthermore, we show that the entropy varies real-analytically in terms of probability measures of constant support, which describe the random walk. This setting applies, in particular, to random walks on virtually free groups.