Concentration of measure and mixing for Markov chains

TL;DR

This paper demonstrates that certain Markov chains on graphs with local dynamics rapidly reach equilibrium and exhibit strong measure concentration, with applications in computer science and statistical mechanics.

Contribution

It establishes conditions under which Markov chains on graphs show rapid mixing and measure concentration, extending understanding of their convergence properties.

Findings

Markov chains on graphs can have rapid convergence to equilibrium.

Strong concentration of measure occurs under certain conditions.

Applications include chains from computer science and statistical mechanics.

Abstract

We consider Markovian models on graphs with local dynamics. We show that, under suitable conditions, such Markov chains exhibit both rapid convergence to equilibrium and strong concentration of measure in the stationary distribution. We illustrate our results with applications to some known chains from computer science and statistical mechanics.