Bounds on Mixing Time for Time-Inhomogeneous Markov Chains

TL;DR

This paper introduces a new concept of mixing time for time-inhomogeneous Markov chains and develops methods to estimate it, applying these to dynamic Erdős-Rényi graphs to determine their mixing times.

Contribution

It proposes a novel mixing time framework for time-inhomogeneous Markov chains and extends existing techniques to analyze dynamic random environments.

Findings

Mixing time for dynamic Erdős-Rényi graphs is O(log(n)) above the connectivity threshold.

Established an almost matching lower bound for the mixing time.

Extended the evolving set method to time-inhomogeneous Markov chains.

Abstract

Mixing of finite time-homogeneous Markov chains is well understood nowadays, with a rich set of techniques to estimate their mixing time. In this paper, we study the mixing time of random walks in dynamic random environments. To that end, we propose a concept of mixing time for time-inhomogeneous Markov chains. We then develop techniques to estimate this mixing time by extending the evolving set method of Morris and Peres (2003). We apply these techniques to study a random walk on a dynamic Erd\H{o}s-R\'enyi graph, proving that the mixing time is $O(\log(n))$ when the graph is well above the connectivity threshold. We also give an almost matching lower bound.