Mixing and cut-off in cycle walks

TL;DR

This paper investigates the cutoff phenomenon and mixing times of random walks on cyclic groups with uniform symmetric generators, providing insights into the transition to stationarity in these Markov chains.

Contribution

It analyzes the cutoff phenomenon for random walks on cyclic groups with uniform symmetric generators, extending understanding of mixing times in these settings.

Findings

Identifies conditions under which cutoff occurs

Provides bounds on mixing times for specific generating sets

Enhances understanding of phase transition in Markov chains

Abstract

Given a sequence $(\mathfrak{X}_i, \mathscr{K}_i)_{i=1}^\infty$ of Markov chains, the cut-off phenomenon describes a period of transition to stationarity which is asymptotically lower order than the mixing time. We study mixing times and the cut-off phenomenon in the total variation metric in the case of random walk on the groups $\mathbb{Z}/p\mathbb{Z}$, $p$ prime, with driving measure uniform on a symmetric generating set $A_p \subset \mathbb{Z}/p\mathbb{Z}$.