Upper bounds on mixing time of finite Markov chains

TL;DR

This paper develops a framework for bounding the mixing times of finite Markov chains with a specific algebraic structure, utilizing graph expansions and loop graphs to derive rational stationary distributions and mixing time bounds.

Contribution

It introduces a novel approach combining algebraic and graph-theoretic methods to estimate mixing times, including a new Markov chain on poset linear extensions with optimal mixing time.

Findings

Provided upper bounds on mixing times for chains with left zero minimal ideal.

Derived rational stationary distributions using loop graphs.

Established an $O(n \,\log n)$ mixing time for a new poset-based Markov chain.

Abstract

We provide a general framework for computing upper bounds on mixing times of finite Markov chains when its minimal ideal is left zero. Our analysis is based on combining results by Brown and Diaconis with our previous work on stationary distributions of finite Markov chains. Stationary distributions can be computed from the Karnofsky--Rhodes and McCammond expansion of the right Cayley graph of the finite semigroup underlying the Markov chain. Using loop graphs, which are planar graphs consisting of a straight line with attached loops, there are rational expressions for the stationary distribution in the probabilities. From these we obtain bounds on the mixing time. In addition, we provide a new Markov chain on linear extension of a poset with $n$ vertices, inspired by but different from the promotion Markov chain of Ayyer, Klee and the last author. The mixing time of this Markov chain is $O(n \log n)$.