# Long-term concentration of measure and cut-off

## TL;DR

This paper develops new concentration inequalities for Markov chains, enabling the analysis of the cut-off phenomenon in both finite and infinite state spaces, with applications to models like Bernoulli-Laplace, disease spread, and supermarket queues.

## Contribution

It introduces generalized concentration of measure inequalities for Markov chains, extending cut-off analysis to infinite state spaces and diverse models.

## Key findings

- Probabilistic proof of cut-off for Bernoulli-Laplace model
- Extended cut-off concept to infinite state space chains
- Concentration results for supermarket model

## Abstract

We present new concentration of measure inequalities for Markov chains, generalising results for chains that are contracting in Wasserstein distance. These are particularly suited to establishing the cut-off phenomenon for suitable chains. We apply our discrete-time inequality to the well-studied Bernoulli-Laplace model of diffusion, and give a probabilistic proof of cut-off, recovering and improving the bounds of Diaconis and Shahshahani. We also extend the notion of cut-off to chains with an infinite state space, and illustrate this in a second example, of a two-host model of disease in continuous time. We give a third example, giving concentration results for the supermarket model, illustrating the full generality and power of our results.