Drift and entropy growth for random walks on groups

TL;DR

This paper explores the asymptotic behavior of drift and entropy in symmetric random walks on groups, revealing new possible growth patterns including near-linear drift and infinitely many entropy growth rates.

Contribution

It introduces novel asymptotic behaviors for drift and entropy in symmetric random walks on groups, expanding understanding of their long-term dynamics.

Findings

Drift can be very close to linear yet sublinear.

Existence of infinitely many entropy growth asymptotics.

New estimates for entropy growth in these random walks.

Abstract

In this paper we consider finitary symmetric random walks on groups. We construct new possible asymptotics for the drift. We show that the drift can be very close to linear ant yet sublinear. We also give estimates for entropy growth of these random walks. In particular, we prove that there exist infinitely many asymptotics of entropy.