Explicit expanders with cutoff phenomena

TL;DR

This paper constructs explicit cubic expander graphs demonstrating the cutoff phenomenon in the convergence of random walks to equilibrium, filling a gap in deterministic examples of such expanders.

Contribution

It provides the first explicit family of cubic expanders with total-variation cutoff, and explores variants with different cutoff behaviors.

Findings

Constructed explicit cubic expanders with cutoff

Demonstrated cutoff from worst-case initial positions

Created variants with no cutoff or cutoff at prescribed times

Abstract

The cutoff phenomenon describes a sharp transition in the convergence of an ergodic finite Markov chain to equilibrium. Of particular interest is understanding this convergence for the simple random walk on a bounded-degree expander graph. The first example of a family of bounded-degree graphs where the random walk exhibits cutoff in total-variation was provided only very recently, when the authors showed this for a typical random regular graph. However, no example was known for an explicit (deterministic) family of expanders with this phenomenon. Here we construct a family of cubic expanders where the random walk from a worst case initial position exhibits total-variation cutoff. Variants of this construction give cubic expanders without cutoff, as well as cubic graphs with cutoff at any prescribed time-point.