### Penn Mathematics Colloquium

Wednesday, October 6, 2021 - 3:45pm

#### Evita Nestoridi

Princeton University

Location

University of Pennsylvania

A4 DRL

Tea at 3:15pm in 4E17 DRL.

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).