Speed exponents of random walks on groups

TL;DR

This paper constructs finitely generated groups where the expected distance of a simple random walk grows at a rate specified by any nice function between n^{3/4} and n, demonstrating precise control over random walk speed.

Contribution

It introduces a method to construct groups with random walk speeds matching any prescribed function within a certain range.

Findings

Constructed groups with random walk speed n^beta for 3/4 <= beta < 1

Demonstrated precise control over the growth rate of random walk distance

Showed that speed can match any nice function up to a constant factor

Abstract

For every 3/4 <= beta < 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point is within a constant factor of n^beta. In fact, the speed can be set precisely to equal any nice prescribed function up to a constant factor.