Long-term concentration of measure and cut-off
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.
Cite
@article{arxiv.1902.00822,
title = {Long-term concentration of measure and cut-off},
author = {Andrew Barbour and Graham Brightwell and Malwina Luczak},
journal= {arXiv preprint arXiv:1902.00822},
year = {2022}
}
Comments
60 pages; substantially revised in response to referee comments; accepted for publication in Stochastic Processes and Applications