This talk discusses the distribution of resonance widths in quantum chaotic systems weakly coupled to the continuum via a finite number $M$ of open channels. In contrast to the standard perturbative treatment of RMT, we do not a-priory assume the resonance widths being small compared to the mean level spacing. We show that to the leading order in weak coupling the perturbative $\chi$-square distribution of the resonance widths (in particular, the Porter-Thomas distribution at $M=1$) should be corrected by a factor related to a certain average of the ratio of square roots of the characteristic polynomial ('spectral determinant') of the underlying RMT Hamiltonian. A simple single-channel expression is obtained that properly approximates the width distribution also at large resonance overlap, where the Porter-Thomas result is no longer applicable. [Based on a joint work with Yan Fyodorov, QMUL.]

Many years have passed since the initial suggestion by Montgomery (1973) that in an appropriate asymptotic limit the zeros of the Riemann zeta function behave statistically like eigenvalues of random matrices, and the subsequent proposal of Katz and Sarnak (1999) that the same is true of families of more general L-functions. While this limiting behaviour is very informative, even more interesting are the intricacies involved in the approach to this limit, the understanding of which allows us to use random matrix theory in novel ways to shed light on major open questions in number theory.

This is joint work with Gaultier Lambert.