Fermionic systems considered with a certain class of potentials at zero temperature and random matrix ensembles as GUE are known to share the same determinantal point process structure, and so in the "large size limit" are governed by universal objects: the Airy and sine kernels. In this talk, we will first review the integrable structure of such objects and their relation with Painlevé equations. Then we will discuss some "finite-temperature" generalizations of these kernels, coming from the same fermionic models when considered at positive temperature, and the generalizations of integrability results. Time permitting, we will discuss other examples of integrable kernels coming from random polymer models that have similar integrability features.

The central question in random matrix theory is the analysis of eigenvalues and eigenvectors, in particular establishing universal properties as pioneered by Wigner in the 1950s. In most cases, this question is studied for a single random matrix. In this talk, I will survey several results concerning joint spectral properties of two random matrices. Our earlier results include an optimal eigenvector decorrelation estimate and analysis of the Loschmidt echo for two differently deformed Wigner matrices, $D_1 + W$ and $D_2 + W$, where $D_1, D_2$ are two different deterministic Hermitian matrices and $W$ is a random Wigner matrix. Moreover, I will explain our newest result, a decorrelation phenomenon for the top eigenvalues of minors of Wigner matrices, which, in particular, allowed us to fully resolve the Paquette-Zeitouni law of fractional logarithm (LFL) in both symmetry classes, as conjectured almost ten years ago. The technical backbone of our results are concentration estimates for products of resolvents of the two random matrices, whose eigenvalues and eigenvectors are studied. These estimates are called multi-resolvent local laws, and for their proof, we developed a novel dynamical method, the zigzag strategy, which is capable of extracting the fine decorrelation effects. Based on several recent joint works with Z. Bao, G. Cipolloni, L. Erdös, and O. Kolupaiev.

Quantum systems with disorder play an important role in physics as regions with localized and delocalized eigenvectors can emerge simultaneously in their spectrum. These regions or phases correspond to isolating or conducting systems, respectively. We proved that the adjacency matrix of the critical Erdős-Rényi graph possesses phases with localized and delocalized eigenvectors when the expected degree is sufficiently small. In a certain regime, there is a sharp transition between these two phases, a so-called mobility edge. We also decompose the solution of the associated time-dependent Schrödinger according to the two phases and determine their contributions. These results have been obtained in joint works with Antti Knowles and Raphael Ducatez.

It is known that the square of the largest eigenvalue of the adjacency matrix of an Erdős-Rényi graph with constant average degree is approximately equal to the maximum degree. In this talk, we prove that the largest eigenvalues can be more precisely determined by small neighborhoods around vertices with maximum degree, and that its eigenvector is exponentially localized. Based on joint work with Theo McKenzie.

The Novikov–Shubin invariant associated to a graph provides information about the accumulation of eigenvalues of the corresponding adjacency matrix close to the origin. A finite value of this invariant indicates a power law behaviour of the eigenvalue density as a function of the distance of the spectral parameter to zero. We provide a complete description of these Novikov–Shubin invariants for the stochastic block model, i.e. for dense random graphs with constant batch sizes and inhomogenous edge densities between the blocks. We treat the case of undirected as well as directed graphs, for which the eigenvalues of the adjacency matrix lie in the complex plane. The invariants depend only on which batches in the graph are connected by non- zero edge densities. We present an explicit finite step algorithm for their computation. For the proof we identify the asymptotic empirical eigenvalue distribution with the spectral density of an element in a non-commutative probability space within operator-valued free probability theory. We determine the spectral density in the bulk regime by solving the associated Dyson equation. Subsequently we infer the singular behaviour of the density close to the origin by deriving an equation for the exponents associated to the power law with which the resolvent entries of the adjacency matrix that correspond to the individual batches diverge to infinity or converge to zero.

Imagine an $NK\times NK$ Hermitian matrix with independent Gaussian entries that is divided into $K\times K$ blocks of size $N\times N$. Let each entry in the $(i, j)$ block have mean zero and variance $S_{ij}\geq0$. Clearly, the distribution of eigenvalues depends on the variance profile encoded by the $K\times K$ matrix $S$. For example, when $S$ contains many zeros, the large-$N$ limit of the eigenvalue density can develop a singularity at the origin, and the singularity degree depends on the pattern of zeros in $S$. I will present results from recent work with Torben Krüger, where we determine the scaling limit of the eigenvalue density at the origin. We compute the density via its Stieltjes transform, for which we derive an integral representation at finite-$N$ using the superbosonization method. Then we derive the large-$N$ scaling limit via a saddle-point approximation.

We will discuss a quantitative Tracy-Widom law for Wigner random matrices, including a class of sparse random matrices which is motivated by the adjacency matrices of the Erdos-Reyni graph model. We will derive a Berry-Esseen type estimate for the fluctuations of the largest eigenvalues of such matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős, Yau, and Yin to prove edge universality, yet on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow induced by a matrix-valued Ornstein-Uhlenbeck process. Precise estimates and cancellation among leading contributions from the third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions. This is joint work with Y. Xu (AMSS) and T. Bucht (KTH).

In this talk, I will discuss universality of the k-point correlation function for non-Hermitian Wigner-type matrices, which are $N\times N$ matrices with centred, independent, and identically distributed entries. I will discuss our new results for complex-valued matrices, established via universality of matrices with a small Gaussian component. We prove that the bulk correlation functions are universal in the large $N$ limit using Householder transformations, supersymmetry, and the Laplace method. Assuming the entries have finite moments and are supported on at least three points, the Gaussian component is removed by the four-moment theorem. I will then give an overview of recent work in the field that substantially extends these results. This is based on joint work with Mohammed Osman.