On the joint behaviour of speed and entropy of random walks on groups

TL;DR

This paper constructs specific groups and random walks where the expected distance and entropy grow at prescribed rates, demonstrating a precise relationship between speed and entropy in group-based random walks.

Contribution

It provides a method to realize any pair of growth functions for speed and entropy within certain bounds on finitely generated groups.

Findings

Speed and entropy can be simultaneously controlled to follow prescribed functions.

Constructed groups exhibit specific growth behaviors for random walks.

The relationship between speed and entropy is characterized within certain bounds.

Abstract

For every $3/4\le \delta, \beta< 1$ satisfying $\delta\leq \beta < \frac{1+\delta}{2}$ we construct a finitely generated group $\Gamma$ and a (symmetric, finitely supported) random walk $X_n$ on $\Gamma$ so that its expected distance from its starting point satisfies $E|X_n|\asymp n^{\beta}$ and its entropy satisfies $H(X_n)\asymp n^\delta$. In fact, the speed and entropy can be set precisely to equal any two nice enough prescribed functions $f,h$ up to a constant factor as long as the functions satisfy the relation $n^{\frac{3}{4}}\leq h(n)\leq f(n)\leq \sqrt{{nh(n)}/{\log (n+1)}}\leq n^\gamma$ for some $\gamma<1$.