Elucidating the cutoff phenomenon

The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity: instead of decaying gradually over time, their distance to equilibrium remains close to the maximal value for a while and suddenly drops to zero as the time parameter reaches a critical threshold known as the mixing time.Discovered four decades ago in the context of card shuffling, this dynamical phase transition has since then been observed in a variety of situations, from random walks on random graphs to high-temperature spin glasses.

It is now believed to be universal among fast-mixing high-dimensional systems.

Yet, the current proofs are case-specific and rely on explicit computations which (i) can only be carried out in oversimplified models and (ii) do not bring any conceptual insight as to why such a sharp transition occurs. Our ambition here is to identify the general conditions that trigger the cutoff phenomenon.

This is one of the biggest challenges in the quantitative analysis of finite Markov chains.We believe that the key is to harness a new information-theoretic statistics called varentropy, whose relevance was recently uncovered by the PI but whose systematic estimation remains entirely to be developed.

Specifically, we propose to elaborate a robust set of analytic and geometric tools to bound varentropy and control its evolution under any Markov semi-group, much like log-Sobolev inequalities do for entropy.

From this, we intend to derive sharp and easily verifiable criteria allowing us to predict cutoff without having to compute mixing times.

If successful, our approach will not only provide a unified explanation for all known instances of the phenomenon, but also confirm its long-predicted occurrence in a number of models of fundamental importance. Emblematic applications include random walks on expanders, interacting particle systems, and MCMC algorithms.