Paul Bourgade

Universality of general

We prove the universality of the -ensembles with convex analytic potentials and for any > 0, i.e. we show that the spacing distributions of log-gases at any inverse temperature coincide with those of the Gaussian -ensembles.

\beta

We prove the edge universality of the beta ensembles for any

Edge Universality of Beta Ensembles

{\beta \ge 1}

The characteristic polynomial of a random unitary matrix: A probabilistic approach

β≥1, provided that the limiting spectrum is supported on a single interval, and the external potential is

Extreme gaps between eigenvalues of random matrices

{\fancyscript{C}^4}

Quantum Chaos, Random Matrix Theory, and the Riemann ζ-function

C4 and regular. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class