A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of card shuffling (Aldous-Diaconis, 1986), this remarkable phenomenon is now rigorously established for many Markov chains. There is however a lack of general theory for proving this phenomenon. In this lecture, in the context of random walks, the speaker will see that strong asymptotic freeness or more generally strong asymptotic convergence of operator algebras can be used to establish cutoff. He will notably illustrate his results for Markov chains whose transition kernel is a non-commutative polynomial in random uniform and independent permutations matrices.
About the speaker
Dr. Charles Bordenave received his PhD from École Normale Supérieure in 2006. He then joined the University of California, Berkeley as a Postdoctoral Fellow and moved to the Institut de Mathématiques of Université de Toulouse in 2007. In 2018, he moved to the Institut de Mathématiques de Marseille and is currently a CNRS Researcher there.
Dr. Bordenave’s research focuses on random matrices, random graphs, stochastic networks, combinatorial optimization and stochastic geometry. He was awarded the Marc Yor Prize by the French Academy of Sciences in 2017.
