Characteristic Polynomials

The aim of this conference is to gather experts in mathematics and physics in the domain of random matrices at IML, to foster research on a timely polymorphic special subject. Random matrix theory is a prominent research topic in Scandinavia, including KTH, Helsinki University and Nordita and this workshop constitutes a follow-up to the IML program on “Random matrices and scaling limit” in the fall 2024.

The characteristic polynomial is a natural random analytic function that encodes the eigenvalues of a matrix and it is an object at the intersection of many research fields in mathematics, such as high-dimensional statistics, spectral theory of random operators, mathematical physics and quantum chaos, but also probabilistic number theory. Historically, characteristic polynomials of Gaussian and unitary random matrices have been used to define the scaling limits of eigenvalues of large random matrices and the corresponding random operators.

Currently, it has been established that the global fluctuations of characteristic polynomials are described by the theory of Gaussian multiplicative chaos (GMC), a canonical class of multi-fractal random objects. This yields new connections between the fields of random matrices and random geometry, in particular two-dimensional field theories. This relationship to GMC has also been successful to study extreme eigenvalue fluctuations, including the maximum of log characteristic polynomials, using techniques from the field of branching processes. These extremal statistics are expected to be universal for log-correlated random landscapes and this led to important conjectures for the maximum of the Riemann ζ-function. In fact, the characteristic polynomial of random unitary matrices is an established probabilistic model for the Riemann ζ-function and its scaling limit, known as the stochastic ζ-function, has been constructed in a recent series of works. Based on this analogy, there has even been recent progress to describe the maximum of the Riemann ζ-function on typical short intervals on the critical line. The problem of universality of fluctuations of characteristic polynomials of Hermitian random matrices is still open, although significant progress has been made recently using dynamical methods.

There are also many important open problems pertaining to non-Hermitian random matrices and two-dimensional Coulomb gas. In these fields, the characteristic polynomial is naturally connected to potential theory via the logarithmic potential and there is a deep connection to the Gaussian free field. The characteristic polynomial viewed as a random analytic function outside the pseudo-spectrum is an object which has been instrumental to study outliers. Finally, there are also connections to multiplicative ergodic theory via transfer matrices and to supersymmetry to study correlation functions.

This workshop aims to further explore the high-dimensional asymptotic questions related to characteristic polynomials for both Hermitian and non-Hermitian models. The goal is to bring together researchers from different fields, such as asymptotic analysis, random operators, statistical physics and probabilistic number theory, to foster new collaborations. We intend to have selected accessible talks and to devote a significant part of the workshop to discuss open problems.