Cutoff for quantum random transpositions

Simeng Wang (Université Paris-Saclay )

Thursday 22nd April 16:00-17:00 Contact the Organizers for Zoom Coordinates

Abstract

The celebrated cutoff phenomenon was first discovered by Diaconis and Shahshahani in 1981 for random transpositions, or intuitively for random “card shuffles” : imagine a deck of N cards spread on a table, randomly select one of them uniformly, and then another one uniformly; if one card is chosen twice, then do nothing; otherwise swap the two cards. For a number of steps, the distribution of permutations of cards stays far apart from stationarity and then it suddenly drops exponentially close to it. In this talk, I will present the similar random walk theory on compact quantum groups, and in particular present a recent analogous result in the setting of quantum random transpositions. I will also discuss the associated asymptotic description of the convergence to equilibrium, called the "limit profile", whose type is different from the classical examples and involves free Poisson distributions emerged from the free probability theory.

This is joint work with Amaury Freslon and Lucas Teyssier (arXiv: 2010.03273).

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