Probability Seminar

Weyl Law in Liouville Quantum Gravity

Nathanaël Berestycki

Can you hear the shape of LQG? We obtain a Weyl law for the eigenvalues of Liouville Brownian motion: the n-th eigenvalue grows linearly with n, with the proportionality constant given by the Liouville area of the domain (times a certain...

On The Cover Time of Random Walks on Graphs

Nathanaël Berestycki

How long does it take for a random walk to cover all the vertices of a graph?


And what is the structure of the uncovered set (the set of points not yet visited by the walk) close to the cover time?


We completely characterize the...

It all began with card shuffling. Diaconis and Shahshahani studied the random transpositions shuffle; pick two cards uniformly at random and swap them. They introduced a Fourier analysis technique to prove that it takes 1/2nlogn steps to shuffle a...