About the speed of random walks on solvable groups

TL;DR

This paper demonstrates that within solvable groups, the speed of random walks can vary continuously, achieving any upper speed exponent between 1/2 and 1, depending on the measure used.

Contribution

It constructs specific solvable groups and measures to realize any desired upper speed exponent in [1/2, 1], showing the range of possible random walk speeds.

Findings

Existence of solvable groups with prescribed speed exponents

Random walks can have upper speed exponents spanning [1/2, 1]

Speed depends on the choice of finitely supported measure

Abstract

We show that for each $\lambda \in [\frac{1}{2}, 1]$, there exists a solvable group and a finitely supported measure such that the associated random walk has upper speed exponent $\lambda$.