Joe P. Chen (Colgate University), The cutoff phenomenon for exclusion processes on graphs with open boundaries

A fundamental question in the study of Markov chains is the convergence to a stationary distribution. As an example, how many shuffles does it take to fully mix a standard deck of 52 cards? Answering this "mixing time" question requires understanding notions of convergence in finite Markov chains, and the spectral theory of Markov matrices.

For certain families of Markov chains, one can exhibit a cutoff phenomenon, in which the chains suddenly converge to stationarity at a well-defined macroscopic time scale.There are by now many known cutoff examples, one of which is the 1D simple exclusion process, where particles perform random walks on the discrete interval {1, 2, . . . , N-1}, subject to the exclusion rule that no two particles may occupy the same vertex at any time. This model can be generalized to include "boundary reservoirs" at the two endpoints 0 and N through which particles can enter, or exit from, the discrete interval. For this boundary-driven exclusion model there are partial (pre-)cutoff results.

In this talk, I will describe a robust method to prove cutoff for boundary-driven exclusion on the d-dimensional integer lattice, for any dimension d. The method involves probabilistic coupling, analysis of martingales, and elementary (!) Fourier analysis that is taught in an undergraduate differential equations course. When suitably generalized, the proof works on non-lattice graphs such as trees and fractals.

This is based on joint work with Milton Jara (IMPA) and Rodrigo Marinho (Tecnico Lisboa).

For more information, please visit https://math.sci.ccny.cuny.edu/seminar/show/7