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 systems. It is now believed to be universal among fast-mixing high-dimensional processes. 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.
The aim of this ERC-funded project is to identify the general conditions that trigger a cutoff. If successful, our approach will not only provide a unified explanation for all known instances of this 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.