A method to derive concentration of measure bounds on Markov chains

TL;DR

This paper discusses a method for deriving concentration of measure bounds on Markov chains, applying it to various models including Kac walks and the asymmetric exclusion process, to establish new probabilistic bounds.

Contribution

It extends a known method to new applications involving spectral gaps and concentration bounds for complex Markov chain models.

Findings

Concentration bounds for Kac walks on SO(n)

Results on longest increasing subsequence length

Application to asymmetric exclusion process

Abstract

We explore a method introduced by Chatterjee and Ledoux in a paper on eigenvalues of principle submatrices. The method provides a tool to prove concentration of measure in cases where there is a Markov chain meeting certain conditions, and where the spectral gap of the chain is known. We provide several additional applications of this method. These applications include results on operator compressions using the Kac walk on $SO(n)$ and a Kac walk coupled to a thermostat, and a concentration of measure result for the length of the longest increasing subsequence of a random walk distributed under the invariant measure for the asymmetric exclusion process.