It was proven by Rider and Virag that the logarithm of the characteristic polynomial of the Ginibre ensemble converges to a logarithmically correlated random field. In this talk we will see how this connection can be established on the level if powers of the characteristic polynomial by proving convergence to Gaussian multiplicative chaos. We consider the range of powers in the \(L^2\) phase.
(Joint work in progress with Paul Bourgade and Guillaume Dubach).

We consider two fundamental percolation models with long-range correlations: The Gaussian free field and (the vacant set) of Random Interlacements. Both models have been the subject of intensive research during the last years and decades, on \(\mathbb Z^d\) as well as on some more general graphs. We investigate some structural percolative properties around their critical parameters, in particular the ubiquity of the infinite components of complementary phases.
This talk is based on joint works with A. Prévost (Köln) and P.-F. Rodriguez (Bures-sur-Yvette).

Finding the (space-height) distribution of the (local) extrema of high-dimensional strongly correlated random fields is a notorious hard problem with many applications. Following Fyodorov and Sommers (2007), we focus on the Gaussian fields with isotropic increments and take the viewpoint of statistical physics. By exploiting various probabilistic symmetries, we rigorously derive the Fyodorov-Sommers formula for the log-partition function in the high-dimensional limit. The formula suggests a rich picture for the distribution of the local extrema akin to the celebrated spherical Sherrington-Kirkpatrick model with mixed p-spin interactions.

For Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner-Dyson-Mehta universality conjecture for the last remaining universality type.

Consider the hypercube as a graph with vertex set \({0,1}^N\) and edges between two vertices if they are only one coordinate flip apart. Choosing independent standard exponentially distributed lengths for all edges and asking how long the shortest directed paths from \((0,..,0)\) to \((1,..,1)\) is defines oriented first passage percolation on the hypercube. We will discuss the conceptual steps needed to answer this question to the precision of extremal process following the two paper series "Oriented first passage percolation in the mean field limit" by Nicola Kistler, Adrien Schertzer and Marius A. Schmidt: arXiv:1804.03117 [math.PR] and arXiv:1808.04598 [math.PR].

We consider the adjacency matrix of the Erdos-Renyi graph \(G(N,p)\) in the supercritical regime \(pN > C \log N\) for some universal constant C. We show that the eigenvalue density is with high probability well approximated by the semicircle law on all spectral scales larger than the typical eigenvalue spacing. We also show that all eigenvectors are completely delocalized with high probability. Both results are optimal in the sense that they are known to be false for \(pN < \log N\). A key ingredient of the proof is a new family of large deviation estimates for multilinear forms of sparse vectors. Joint work with Yukun He and Matteo Marcozzi.

The goal of this talk is to explain how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations. These are known as “rigidity estimates” in the literature and they play a crucial role in the proof of universality. I will review some of the current results on eigenvalues’ fluctuations and present a new approach which relies on the theory of Gaussian Multiplicative Chaos and leads to optimal rigidity estimates for the Gaussian Unitary Ensemble. I will also mention how it is also deduce a central limit theorem from our proof. This is joint work with Tom Claeys, Benjamin Fahs and Christian Webb.

Uniform integer-valued Lipschitz functions on a finite domain of the triangular lattice are shown to have variations of logarithmic order in the radius of the domain. The level lines of such functions form a loop \(O(2)\) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop \(O(2)\) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at 0; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop \(O(2)\) model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model. Based on joint work with Alexander Glazman.