Long-term concentration of measure and cut-off

Andrew Barbour; Graham Brightwell; Malwina Luczak

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arXiv · 1902.00822

Long-term concentration of measure and cut-off

Andrew Barbour·Graham Brightwell·Malwina Luczak

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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.

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Andrew Barbour, Graham Brightwell, Malwina Luczak. 2019-02-03. Long-term concentration of measure and cut-off. https://arxiv.org/abs/1902.00822

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