Title & Abstract

Abstract: The Liouville quantum gravity measure is a properly normalized exponential of 2d log-correlated fields, such as the Gaussian free field. It is the volume form for the scaling limit of random planar maps and numerous statistical physics models. I will explain how this random measure naturally appears in random matrix theory either in space time from random matrix dynamics, or in space from the characteristic polynomial of random normal matrices. Central to these results are joint moments of characteristic polynomials for the Dyson Brownian motion, or for Ginibre matrices, which are analogues of Fisher Hartwig asymptotics in dimension 2.

Abstract: Non-Hermitian random matrices provide a useful framework for understanding universal characteristics of dissipative quantum chaotic systems with loss or gain. We consider a model of two such system represented by two independent $N\times N$ complex Ginibre matrices interacting via a deterministic matrix $c{\bf 1}_N$, where $c$ is the complex coupling parameter whose magnitude $|c|$ controls the interaction strength. We characterize quantitatively how the eigenvectors of the whole system, initially localized in one of the individual subsystems for $|c|=0$, eventually spread over the full system with growing interaction strength. The resulting asymptotic formula describing such spread in the limit $N\to \infty$ is very explicit and provides a full picture of the gradual ergodization of eigenvectors as a function of the coupling parameter $|c|$ in the whole transition regime.

The presentation will be based on the joint work with Margherita Disertori: arXiv:2604.23708.

Abstract: Bogomolny, Bohigas and Leboeuf (1992) argued that the variance profile of the coefficients of high-degree random polynomials governs a crystal-to-liquid transition in the distribution of their zeros. 

 In this talk I will describe this transition rigorously in the setting of random polynomials with regularly varying coefficients. In particular, I will explain how the local statistics of zeros near the unit circle exhibit a phase transition at a critical exponent, separating a crystalline regime with lattice-like zero patterns from a liquid regime described by zeros of random entire functions. I will also discuss universality properties of the limiting point processes across these regimes. 

 This is joint work with Zakhar Kabluchko and Oleksandr Marynych.

Abstract: Potential energies in disordered solid materials will always exhibit some degree of spatial correlations. Yet the majority of studies of electron localization in disordered systems is formulated using independent random on-site potentials. I will present a method that allows one to obtain spectra of Random Schrödinger Operators (RSOs) on random graphs with correlated random on-site potentials. The study is in part motivated by the recent solution of the level set percolation problem for multivariate Gaussians on complex networks. Among other things, I will look at the question whether, for a state of a system to be extended, the set of sites G_E = {i; V_i(x) < E} on which the energy of the state exceeds the potential energy should itself be percolating and thus extensive. 

 Abstract: Since the celebrated  1970 lesson of Ginibre in Les Houches, it has been formally established that, under broad hypotheses, a canonical ensemble of N indistinguishable bosons in a bounded domain at thermal equilibrium can be modeled as a point process. Its distribution is called  the Feynman-Kac representation of the Bose gas and can be interpreted as a model of random spatial permutations. In a joint work with  Guillaume Bellot and David Dereudre (Université de Lille),  we construct a thermodynamic limit for the grand canonical version of this Bose gas  with superstable interaction, at any  temperature and chemical potential. Our infinite volume model is naturally a distribution over configurations of finite loops and possibly interlacements. We prove the limiting process to solve a new class of DLR equations involing random permutations and Brownian paths. 

Abstract: RMT and point processes have been applied to the world at large in a myriad of ways. I here dive in to the world of where birds of prey like to build their nests and how territorial behaviour can be estimated by comparison to point processes, specifically the nearest-neighbour spacing distribution of a 2D Coulomb Gas. In this talk, I want to answer the questions

• How to make good numerical estimates of the nearest-neighbour spacing of the Coulomb gas where no analytic results are known?

• How to unfold the 2D-spectrum in a way that preserves local distances?

• What alternative effects could explain the territorial effects we see?

• What are the differences if we look at one species internally versus inter-species interaction? (We have data for buzzards, goshawks, and eagle owls.)

This is talk is based on the following articles

• arXiv:2310.20670

• arXiv:2003.09204

possibly with some results from

• arXiv:2603.28457

• arXiv:2112.12624

• arXiv:1910.03520

as well.

Abstract: The distribution P(r) of ratios of consecutive level spacings, introduced by Oganesyan and Huse (2007), has a key advantage over conventional local spectral statistics: it is insensitive to the global density of states. In this talk, I determine P(r) for the Circular Unitary Ensemble (CUE), as well as the distribution of ratios of the two smallest eigenvalues in Jacobi Unitary Ensembles (JUE), at finite N. These results are obtained from the Tracy–Widom systems of differential equations governing the Janossy densities of the respective ensembles.

 Remarkably, the leading corrections to the large-N limit are universally of order O(1/N^4), reflecting nontrivial cancellations of the O(1/N^2) contributions in the corresponding kernels. This observation can be exploited to quantify deviations in the statistics of zeros of the Riemann zeta function and various L-functions from the predictions of (non-)chiral circular ensembles.

 Based on: 10.1093/ptep/ptag006

Abstract: The conjectured three generic local bulk statistics amongst all non-Hermitian random matrix symmetry classes have recently been extended to three generic local edge statistics. The three simplest representatives of these universality classes are given by the Gaussian ensembles of complex Ginibre, complex symmetric and complex self-dual matrices, denoted by class A, AI$^\dag$ and AII$^\dag$.

In the first part, I present analytical results on the complex spacing ratio in class A, at finite matrix size $N$. When specifying to the elliptic Ginibre ensemble, I present a parameter-dependent $N=3$ surmise for the complex spacing ratio, interpolating to that of the Gaussian unitary ensemble (GUE), where such a surmise is very accurate.

 In the second numerical part, I compare complex spacing ratios, its moments, and NN spacing distributions for all three ensembles with that of uncorrelated points, the two-dimensional (2D) Poisson process, both in the bulk and at the edge. We find indications that the complex spacing ratio does not fully unfold the local statistics at the edge.

 This is joint work with Gernot Akemann, Georg Angermann, Noah Aygün, Adam Mielke, Christoph Raitzig, and Tobias Winkler.

Abstract: In this talk,  I will introduce an infinitely-many neutral allelic model of population genetics where all alleles are  divided into a finite number of classes, and each class is characterized by its own mutation rate.  For this model, the allelic composition of a sample taken from a very large population of genes is characterized by a random matrix, and  the   problem is to describe  the joint distribution of the matrix entries. The answer is given by a new generalization of the classical Ewens sampling formula called the refined Ewens sampling formula. I will discuss a Poisson approximation for the refined Ewens sampling formula and explain its derivation. As an application, I will present limit theorems for the numbers of alleles in different asymptotic regimes.

Abstract: Before the work of Gernot on the QCD Dirac spectrum, the concept of universality in random matrix theory was largely unknown in the  Quantum Chromodynamics community. However, universality was understood in a different way, as the  uniqueness of the low-energy effective action,  which is determined by symmetries and their spontaneous breaking. We will review these ideas and discuss their connections and implications.