Minimal growth harmonic functions on lamplighter groups

TL;DR

This paper investigates the minimal growth rates of harmonic functions on lamplighter groups, revealing that certain lamplighter groups do not admit harmonic functions with growth below specific thresholds, impacting related probabilistic theorems.

Contribution

It establishes new bounds on the minimal growth of harmonic functions on lamplighter groups, extending understanding of harmonic analysis on these groups.

Findings

$( ext{Z}/2) times ext{Z}$ has no sublinear harmonic functions.

$( ext{Z}/2) times ext{Z}^2$ has no sublogarithmic harmonic functions.

Repeated wreath products also lack harmonic functions with growth below certain thresholds.

Abstract

We study the minimal possible growth of harmonic functions on lamplighters. We find that $(\mathbb{Z}/2)\wr \mathbb{Z}$ has no sublinear harmonic functions, $(\mathbb{Z}/2)\wr \mathbb{Z}^2$ has no sublogarithmic harmonic functions, and neither has the repeated wreath product $(\dotsb(\mathbb{Z}/2\wr\mathbb{Z}^2)\wr\mathbb{Z}^2)\wr\dotsb\wr\mathbb{Z}^2$. These results have implications on attempts to quantify the Derriennic-Kaimanovich-Vershik theorem.