The Coulomb Gas and Allied Topics
Monday, May 22 at 13.15-14.15
Speaker: Felipe Marceca (King’s College London)
Title: Discrepancy estimates for the planar Coulomb gas at low temperatures
In this talk we study the planar Coulomb gas for an inverse temperature that grows at least logarithmically with respect to the number of particles. We show that almost surely, the discrepancy between the number of particles in any microscopic region and its expected value is, up to log factors, of the order of the perimeter of the inspected region. The estimates, valid both in the bulk and at the boundary, are achieved by building on recent results on equidistribution at low temperatures and providing refined spectral asymptotics for certain Toeplitz operators on the range of the erfc-kernel. Joint work with Jos´e Luis Romero.
Tuesday, May 23 at 13.15-14.15
Speaker: Yacin Ameur (Lund University)
Title: The two-dimensional Coulomb gas: fluctuations through a spectral gap
The talk will focus around some new results relating to fluctuations of smooth linear statistics of two-dimensional Coulomb gas ensembles, where the droplet is disconnected, i.e., it has some “spectral gaps”. The particles which fall near the boundary of the gap come with an additional uncertainty, since there are several disjoint boundary components to choose from. For a class of centrosymmetric model ensembles, we quantify this uncertainty in terms of the Jacobi theta function as well as a discrete Gaussian distribution. If time permits, I will also discuss a related large n expansion of the partition function. Joint work with Christophe Charlier and Joakim Cronvall (both from Lund University)
Tuesday, May 23 at 15.15-16.15
Speaker: Leslie Molag (University of Sussex)
Title: Edge universality of random normal matrices generalizing to higher dimensional DPPs
As recently proved in generality by Hedenmalm and Wennman, it is a universal behavior of complex random normal matrix models that one finds a complementary error function behavior at the boundary (also called edge) of the droplet as the matrix size increases. Such behavior is seen both in the density of the eigenvalues, and the correlation kernel, where the Faddeeva plasma kernel emerges. These results are neatly expressed with the help of the outward unit normal vector on the edge. We prove that such universal behaviors transcend this class of random normal matrices, being also valid in higher dimensional determinantal point processes, defined by a Bergman kernel on Cd . The models under consideration concern higher dimensional generalizations of the determinantal point processes describing the eigenvalues of the complex Ginibre ensemble and the complex elliptic Ginibre ensemble. These models describe a system of particles in Cd with mutual repulsion, that are confined to the origin by an external field V(z) = |z|2 − τRe(z2/1 + . . . + z2/d ), where 0 ≤ τ < 1. Their correlation kernels Kn(z, w) can be constructed with multivariate orthogonal polynomials P(z1, . . . , zd), that satisfy orthogonality relations with respect to a multivariate weight W(z) = e−V(z) . Their average density of particles converges to a uniform law on a 2d-dimensional ellipsoidal region. It is on the hyperellipsoid bounding this region that we find a complementary error function behavior and the Faddeeva plasma kernel. To the best of our knowledge, this is the first instance of the Faddeeva plasma kernel emerging in a higher dimensional model. The results provide evidence for a possible edge universality theorem for determinantal point processes on Cd.
Wednesday, May 24 at 13.15-14.15
Speaker: Meng Yang (University of Copenhagen)
Title: Normal Matrix Models with Merging Singularity
We study the normal matrix model, also known as the two-dimensional one-component plasma at a specific temperature, with merging singularity. As the number n of particles tends to infinity we obtain the local correlation kernel at the singularity, which is related to the solution of the Painlev´e-II equation. The two main tools we use are Riemann-Hilbert problems and the generalized Christoffel-Darboux identity. The correlation kernel exhibits a novel anisotropic scaling behavior, where the corresponding spacing scale of particles is n−1/3 in the direction of merging and n−1/2 in the perpendicular direction.
Wednesday, May 24 at 15.15-16.15
Speaker: Gaultier Lambert (KTH Stockholm)
Title: Gaussian multiplicative chaos and random matrices
The fluctuations of the eigenvalues of random matrices can be described using Gaussian log-correlated fields by considering for instance the eigenvalue counting function and the log. characteristic polynomial (or 2d Coulomb potential). This description holds for both Hermitian (GUE type) and non-Hermitian matrices (Ginibre type) and the fluctuations can be related to the 2d Gaussian free field. In this talk, I will review these connections and explain how this relate to the theory of Gaussian multiplicative chaos. This framework can be used to describe the extreme values or "thick points" of these log-correlated fields and deduce sharp rigidity results for the eigenvalues of the underlying random matrices. I will try to report on some recent results in this field, including some joint works with Claeys, Fahs, Webb, Junnila, Webb and Leblé, Zeitouni.
Tid: 2023-05-22 13:15 till 2023-05-26 16:15
Room MH:332B, Centre for Mathematical Sciences
yacin [dot] ameur [at] math [dot] lu [dot] se