Tag - Random matrices

Alice Guionnet: About Universality Classes in Random Matrix Theory

25th March, 2024

Wigner’s surmise states that the spectrum of the Hamiltonian of heavy nuclei is distributed like that of a large random matrix. Since it was proposed by Wigner in 1956, the eigenvalue distribution of large random matrices has been used as a toy model to study the distribution of more complex mathematical objects such as random tiles or the longest increasing subsequence of a random perturbation. However, this universality phenomenon generally concerns distributions derived from Gaussian matrices, known as the Gaussian ensembles. In this talk, we will discuss more general universality classes that appear in the theory of random matrices.

Thomas Krajewski: Loop vertex expansion for random matrices with higher order interactions

14th June, 2023

The loop vertex expansion is an alternative to the standard Feynman graph expansion which trades the latter for a convergent expansion over trees. In this talk, we present the general framework and apply it to some random matrix models. As a byproduct, we establish analyticity in the coupling in a domain independent of the size of the matrix, as well as Borel summability.

Paul Bourgade: Random matrices, the Riemann zeta function and branching processes II

8th April, 2023

Random matrix theory is a powerful tool for prediction in analytic number theory. Through this random matrix analogy, Fyodorov, Hiary and Keating conjectured very precisely the typical values of the Riemann zeta function in short intervals of the critical line, in particular their maximum. Their prediction relied on techniques from statistical mechanics such as the replica method, giving extreme values in disordered systems. Recent rigorous progress has exploited underlying branching structures instead, both for random characteristic polynomials and L-functions.

Paul Bourgade: Random matrices, the Riemann zeta function and branching processes I

8th April, 2023

Random matrix theory is a powerful tool for prediction in analytic number theory. Through this random matrix analogy, Fyodorov, Hiary and Keating conjectured very precisely the typical values of the Riemann zeta function in short intervals of the critical line, in particular their maximum. Their prediction relied on techniques from statistical mechanics such as the replica method, giving extreme values in disordered systems. Recent rigorous progress has exploited underlying branching structures instead, both for random characteristic polynomials and L-functions.