Three steps mixing for general random walks on the hypercube at criticality

TL;DR

This paper introduces a class of random walks on the hypercube, identifying a critical range that enables near-perfect mixing in at most three steps, with some cases achieving mixing in exactly two steps.

Contribution

It provides a new framework for analyzing mixing times of hypercube random walks and identifies a critical range for long-range processes that achieve rapid mixing.

Findings

Existence of a critical range for long-range walks enabling rapid mixing.

Almost-perfect mixing occurs in at most three steps, sometimes in exactly two.

Total variation distance decreases geometrically with hypercube dimension.

Abstract

We introduce a general class of random walks on the $N$-hypercube, study cut-off for the mixing time, and provide several types of representation for the transition probabilities. We observe that for a sub-class of these processes with long range (i.e. non-local) there exists a critical value of the range that allows an "almost-perfect" mixing in at most three steps. In other words, the total variation distance between the three steps transition and the stationary distribution decreases geometrically in $N$, which is the dimension of the hypercube. In some cases, the walk mixes almost-perfectly in exactly two steps. Notice that a well-known result (Theorem 1 in Diaconis and Shahshahani (1986)) shows that there exist no random walk on Abelian groups (such as the hypercube) which mixes perfectly in exactly two steps.