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Wednesday programme

Room B3.03

Morning Session Chair: Gregory Schehr

Afternoon Session Chair: Nina Snaith

Time Speaker Title Abstract
09:00-09:50 Igor Krasovsky Transition asymptotics for Toeplitz and Fredholm determinants We will discuss recent results on the asymptotic behaviour of certain Toeplitz and Fredholm determinants appearing in the theory of random matrices and related areas focusing on the situation where a transition between $2$ different asymptotic regimes takes place. The talk is based on joint works with Th. Bothner, T. Claeys, P. Deift, and A. Its.
09:50-10:40 Mihail Poplavskyi Asymptotic behaviour of the largest real eigenvalue for the real Ginibre ensemble We discuss the classical real Ginibre ensemble and the distribution of its biggest real eigenvalue. The distribution was recently obtained in the paper of B. Rider and C. Sinclair by using the classical method coming from C. Tracy and H. Widom papers. We present another elegant way, based on integrability of the model and skew-orthogonal polynomials technique, to obtain the distribution in a much shorter form. We also show how to simplify older answer to a new one by introducing Gaussian Brownian motion. Giving a pure probabilistic interpretation to the answer we are able to perform an asymptotic analysis of the distribution and find its tails. The talk is based on a joint project with R. Tribe and O. Zaboronski.
10:40-11:00 Coffee in the Common Room
11:00-11:50 Dmitry Savin Resonance width distribution beyond Porter-Thomas

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that properly approximates the width distribution also at large resonance overlap, where the Porter-Thomas result is no longer applicable.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.]

11:50-12:40 Nina Snaith Combining random matrix theory and number theory

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.

12:40-14:10 Lunch in the Common Room
14:10-15:00 Nick Simm Global fluctuations for the real component of the Ginibre orthogonal ensemble Let $G$ be an $N\times N$ real matrix of independent identically distributed standard Gaussians. The eigenvalues of $G$ are known to form a two-component system of purely real and complex conjugated points. The real component of this system has interesting applications, from annihilating Brownian motions in probability theory to superconducting level crossings in quantum transport. I will discuss recent progress on our understanding of the real component, including a proof that linear statistics of the real eigenvalues fluctuate on a scale $N^{1/4}$ and satisfy a central limit theorem as $N$ goes to infinity.
15:00-16:00 Tea in the Common Room
16:00-16:50 Elliot Paquette The law of large numbers for the maximum of the log-characteristic polynomial associated to GUE We give a proof that the maximum of the centered log-characteristic polynomial of an $NxN$ GUE matrix is $\log N + o(\log N)$ with high probability. This confirms the first term in a conjecture by Fyodorov and Simm. Moreover, we prove a general theorem about almost Gaussian fields that should be applicable to showing the law of large numbers for the log characteristic polynomial of random matrices.

This is joint work with Gaultier Lambert.

18:00-20:00 Dinner at Radcliffe