Rapid mixing in Markov chains

TL;DR

This paper discusses the rapid mixing of Markov chains used for sampling in complex counting problems, demonstrating polynomial-time convergence using conductance and other techniques across various applications.

Contribution

It provides a proof that certain Markov chains converge quickly to their steady state, facilitating efficient sampling for complex counting problems.

Findings

Markov chains for counting problems mix rapidly in polynomial time

Conductance is a key tool for analyzing convergence

Applications include permanent estimation, partition functions, and volume computation

Abstract

A wide class of ``counting'' problems have been studied in Computer Science. Three typical examples are the estimation of - (i) the permanent of an $n\times n$ 0-1 matrix, (ii) the partition function of certain $n-$ particle Statistical Mechanics systems and (iii) the volume of an $n-$ dimensional convex set. These problems can be reduced to sampling from the steady state distribution of implicitly defined Markov Chains with exponential (in $n$) number of states. The focus of this talk is the proof that such Markov Chains converge to the steady state fast (in time polynomial in $n$). A combinatorial quantity called conductance is used for this purpose. There are other techniques as well which we briefly outline. We then illustrate on the three examples and briefly mention other examples.