The cutoff phenomenon is a sharp transition in the convergence of high-dimensional Markov chains to equilibrium: the total variation distance remains close to 1 for a long time and then rapidly decreases to almost 0 over a much shorter time window.

 

It was initially discovered in the context of card shuffling by Diaconis and Shahshahani, and since then observed in a variety of different models. In spite of its ubiquity, it is still largely unexplained, and most proofs are model-specific.

In this talk, we discuss a high-level approach to establishing cutoff based on transport inequalities, and we illustrate it on a popular algorithm known as the proximal sampler, when the target measure on R^d is log-concave.

 

Based on joint work with Justin Salez.