Related papers: Rising GUE Eigenvalue Process from a Fixed Level
Multitime quantum correlation functions are central objects in physical science, offering a direct link between experimental observables and the dynamics of an underlying model. While experiments such as 2D spectroscopy and quantum control…
Proofs of convergence of adaptive finite element methods for the approximation of eigenvalues and eigenfunctions of linear elliptic problems have been given in a several recent papers. A key step in establishing such results for multiple…
Density estimation represents one of the most successful applications of Bayesian nonparametrics. In particular, Dirichlet process mixtures of normals are the gold standard for density estimation and their asymptotic properties have been…
We consider the spectral statistics of the superposition of a random diagonal matrix and a GUE matrix. By means of two alternative superanalytic approaches, the coset method and the graded eigenvalue method, we derive the two-level…
The solution to a multivariate linear Stochastic Differential Equation (SDE) with constant initial state is well known to be a Gaussian Markov process, but its covariance kernel involves the solution to an integral equation in the general…
The rational points of a smooth curve $X$ over a number field $k$ map to the set of augmentations of the associated motivic algebra. An expectation, related to Kim's conjecture, is that for $X$ hyperbolic, the set of augmentations which…
We prove that the linear statistics of the eigenvalues of a Wigner matrix converge to a universal Gaussian process on all mesoscopic spectral scales, i.e. scales larger than the typical eigenvalue spacing and smaller than the global extent…
We study the universality of the local eigenvalue statistics of Gaussian divisible Hermitian Wigner matrices. These random matrices are obtained by adding an independent GUE matrix to an Hermitian random matrix with independent elements, a…
We consider an $N$ by $N$ real or complex generalized Wigner matrix $H_N$, whose entries are independent centered random variables with uniformly bounded moments. We assume that the variance profile, $s_{ij}:=\mathbb{E} |H_{ij}|^2$,…
This paper is concerned with the asymptotic distribution of the largest eigenvalues for some nonlinear random matrix ensemble stemming from the study of neural networks. More precisely we consider $M= \frac{1}{m} YY^\top$ with $Y=f(WX)$…
This paper is concerned with error estimates of the fully discrete generalized finite element method (GFEM) with optimal local approximation spaces for solving elliptic problems with heterogeneous coefficients. The local approximation…
We consider the determinantal point process with the confluent hypergeometric kernel. This process is a universal point process in random matrix theory and describes the distribution of eigenvalues of large random Hermitian matrices near…
We study three instances of log-correlated processes on the interval: the logarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, the Gaussian log-correlated potential in presence of edge charges, and the Fractional…
We investigate the overlap matrix between the eigenvectors of a Wigner matrix $H_{N+K}$ of size $(N+K)\times(N+K)$ and those of its principal minor $H_N$ of size $N\times N$, for both the real symmetric ($\beta=1$) and complex Hermitian…
We consider a nonlinear singularly perturbed PDE leaning on a complex perturbation parameter $\epsilon$. The problem possesses an irregular singularity in time at the origin and involves a set of so-called moving turning points merging to 0…
Level curvature is a measure of sensitivity of energy levels of a disordered/chaotic system to perturbations. In the bulk of the spectrum Random Matrix Theory predicts the probability distributions of level curvatures to be given by…
We prove eigenvalue processes from dynamical random matrix theory including Dyson Brownian motion, Wishart process, and Dynkin's Brownian motion of ellipsoids are results of projecting Brownian motion through Riemannian submersions induced…
We describe a new universality class for unitary invariant random matrix ensembles. It arises in the double scaling limit of ensembles of random $n \times n$ Hermitian matrices $Z_{n,N}^{-1} |\det M|^{2\alpha} e^{-N \Tr V(M)} dM$ with…
The real Ginibre ensemble consists of random $N \times N$ matrices formed from i.i.d. standard Gaussian entries. By using the method of skew orthogonal polynomials, the general $n$-point correlations for the real eigenvalues, and for the…
In this paper, we survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications,…