Cutoff for lamplighter chains on fractals

TL;DR

This paper investigates the mixing time behavior of lamplighter random walks on fractal graphs, revealing a sharp cutoff phenomenon in transient cases and its absence in recurrent cases based on spectral dimension.

Contribution

It establishes a clear dichotomy in cutoff behavior of lamplighter chains on fractals depending on the spectral dimension of the underlying graph.

Findings

Sharp cutoff occurs for transient fractal graphs with spectral dimension > 2.

No cutoff occurs for recurrent fractal graphs with spectral dimension < 2.

The spectral dimension determines the mixing time behavior of lamplighter chains.

Abstract

We show that the total-variation mixing time of the lamplighter random walk on fractal graphs exhibit sharp cutoff when the underlying graph is transient (namely of spectral dimension greater than two). In contrast, we show that such cutoff can not occur for strongly recurrent underlying graphs (i.e. of spectral dimension less than two).