School of Mathematics

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...