Dull cut off for circulants

TL;DR

This paper demonstrates that symmetric simple random walks on Cayley graphs of Abelian groups with bounded generators do not exhibit the sharp cut off phenomenon in total variation convergence, aligning with Peres' conjecture.

Contribution

It provides the first examples of bounded degree, non-expanding graphs where sharp cut off does not occur, supporting conjectures linking spectral gap and mixing time.

Findings

No sharp cut off in bounded degree, non-expanding Cayley graphs.

Supports Peres' conjecture relating spectral gap and cut off.

Contrasts with known cases of hypercubes and regular graphs with expansion.

Abstract

Families of symmetric simple random walks on Cayley graphs of Abelian groups with a bound on the number of generators are shown to never have sharp cut off in the sense of [1], [3], or [5]. Here convergence to the stationary distribution is measured in the total variation norm. This is a situation of bounded degree and no expansion. Sharp cut off or the cut off phenomenon has been shown to occur in families such as random walks on a hypercube [1] in which the degree is unbounded as well as on a random regular graph where the degree is fixed, but there is expansion [4]. Our examples agree with Peres' conjecture in [3] relating sharp cut off, spectral gap, and mixing time.