Brandeis-Harvard-MIT-Northeastern Joint Mathematics Colloquium

Friday, October 22, 2021 at 4:30pm to 6:00pm

MIT, 2-190 77 Massachusetts Avenue, Cambridge, MA 02139

Speaker:   Evita Nestoridi (Princeton University)

Title:  Spectral techniques in Markov chain mixing

Abstract:  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 $\frac{1}{2} n log n$ 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).

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