The moduli space of twisted Laplacians and random matrix theory
Vortrag von Prof. Dr. Jens Marklof
Datum: 04.12.24 Zeit: 13.30 - 14.30 Raum: Y27H28
One of the long-standing conjectures in quantum chaos is that the spectral statistics of quantum systems with chaotic classical limit are governed by random matrix theory. Despite convincing heuristics, there is currently not a single example where this phenomenon can be established rigorously. Rudnick recently proved that the spectral number variance for the Laplacian of a large compact hyperbolic surface converges, in a certain scaling limit and when averaged with respect to the Weil-Petersson measure on moduli space, to the number variance of the Gaussian Orthogonal Ensemble of random matrix theory. In this lecture we will review Rudnick's approach and extend it to explain the emergence of the Gaussian Unitary Ensemble for twisted Laplacians (which break time-reversal symmetry) and to the Gaussian Symplectic Ensemble for Dirac operators. This addresses a question of Naud, who obtained analogous results for twisted Laplacians on high genus random covers of a fixed compact surface. This lecture is based on joint work with Laura Monk (Bristol).