Positive speed for high-degree automaton groups

TL;DR

This paper demonstrates that random walks on high-degree mother groups exhibit positive speed, contrasting with lower-degree cases, by analyzing resistance properties in fractal mother graphs.

Contribution

It establishes positive speed for random walks on mother groups of degree at least 3, using resistance analysis in fractal graphs, which is a novel approach.

Findings

Random walks on mother groups of degree ≥ 3 have positive speed.

Resistance in fractal mother graphs is bounded, leading to transience.

Infinite mother graphs are shown to be transient.

Abstract

Mother groups are the basic building blocks for polynomial automaton groups. We show that, in contrast with mother groups of degree 0 or 1, any bounded, symmetric, generating random walk on the mother groups of degree at least 3 has positive speed. The proof is based on an analysis of resistance in fractal mother graphs. We give upper bounds on resistances in these graphs, and show that infinite versions are tran- sient.