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Spectral techniques in Markov chain mixing

Evita Nestoridi

How many steps does it take to shuffle a deck of n cards, if at each step we pick two cards uniformly at random and swap them? Diaconis and Shahshahani proved that 12nlogn steps are necessary and sufficient to mix the deck. Using the representation theory of the symmetric group, they proved that this random transpositions card shuffle exhibits a sharp transition from being unshuffled to being very well shuffled. This is called the cutoff phenomenon. In this talk, I will explain how to use the spectral information of a Markov chain to study cutoff. As an application, I will briefly discuss the random-to-random card shuffle (joint with M. Bernstein) and the non-backtracking random walk on Ramanujan graphs (joint with P. Sarnak).


Simons- Math Colloquium