Asymptotic Word Length of Random Walks on HNN Extensions

TL;DR

This paper studies the asymptotic behavior of transient random walks on HNN extensions, establishing the existence of the rate of escape, a central limit theorem, and the real-analytic nature of the escape rate as a function of walk parameters.

Contribution

It proves the existence of the rate of escape, derives a central limit theorem, and shows the real-analytic dependence of the escape rate on parameters for random walks on HNN extensions.

Findings

Rate of escape exists for the considered random walks.

A central limit theorem is established for the generalised word length.

The escape rate varies as a real-analytic function with respect to parameters.

Abstract

In this article we consider transient random walks on HNN extensions of finitely generated groups. We prove that the rate of escape w.r.t. some generalised word length exists. Moreover, a central limit theorem with respect to the generalised word length is derived. Finally, we show that the rate of escape, which can be regarded as a function in the finitely many parameters which describe the random walk, behaves as a real-analytic function in terms of probability measures of constant support.