Remarks on cutoff phenomena for random walks on Hamming Schemes

TL;DR

This paper investigates the cutoff phenomenon for simple random walks on Hamming schemes, providing explicit bounds for total variation distances to better understand the mixing times.

Contribution

It introduces simple majorant and sharp minorant functions for total variation distances in the context of random walks on Hamming schemes with q ≥ 3.

Findings

Established cutoff phenomenon for q ≥ 3

Derived explicit bounds for total variation distances

Provided simple functions to estimate mixing times

Abstract

The sequence of the simple random walks on Hamming schemes $\{H(n, q)\}_{n=1}^{\infty}$ has a cutoff phenomenon for each integer $q$ greater than or equal to $3$. In this paper, for the sequence of simple random walks on Hamming schemes $\{H(n, q)\}_{n=1}^{\infty}$ with $q\geq 3$, we give a simple majorant and a sharp minorant function for total variance distances between transition distributions and stationary distributions.