Related papers: Eigenvalue distribution of large random matrices, …
A one-parameter family of point processes describing the distribution of the critical points of the characteristic polynomial of large random Hermitian matrices on the scale of mean spacing is investigated. Conditionally on the Riemann…
We present a random-matrix realization of a two-dimensional percolation model with the occupation probability $p$. We find that the behavior of the model is governed by the two first extreme eigenvalues. While the second extreme eigenvalue…
In the last decade there has been increasing interest in the fields of random matrices, interacting particle systems, stochastic growth models, and the connections between these areas. For instance, several objects appearing in the limit of…
We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay…
We study a new class of matrix models, formulated on a lattice. On each site are $N$ states with random energies governed by a Gaussian random matrix Hamiltonian. The states on different sites are coupled randomly. We calculate the density…
We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. We consider one-cut regular polynomial potentials and a large class of multiplicative statistics. We show that in the large matrix…
Random matrix theory allows for the deduction of stability criteria for complex systems using only a summary knowledge of the statistics of the interactions between components. As such, results like the well-known elliptical law are…
Dyson's short-distance universality of the correlation functions implies the universality of P(s), the level-spacing distribution. We first briefly review how this property is understood for unitary invariant ensembles and consider next a…
We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the…
What is the connection of random matrices with integrable systems? Is this connection really useful? Introducing apprpriate times in the distribution of the ensemble of matrices, one shows that the corresponding distribution of the…
The paper discusses progress in understanding statistical properties of complex eigenvalues (and corresponding eigenvectors) of weakly non-unitary and non-Hermitian random matrices. Ensembles of this type emerge in various physical…
Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers \cite{Adler:2009a, Daems:2007} and sample covariance matrices \cite{Baik:2005}. We consider the case when the $n\times n$…
This is the second part of a study of the limiting distributions of the top eigenvalues of a Hermitian matrix model with spiked external source under a general external potential. The case when the external source is of rank one was…
We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These…
Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced…
The theory of random matrices with eigenvalues distributed in the complex plane and more general "beta-ensembles" (logarithmic gases in 2D) is reviewed. The distribution and correlations of the eigenvalues are investigated in the large N…
We derive a common mathematical formulation for the eigenfunction statistics of Hermitian operators, represented by a multi-parametric probability density. The system-information in the formulation enters through two parameters only,…
For a large class of symmetric random matrices with correlated entries, selected from stationary random fields of centered and square integrable variables, we show that the limiting distribution of eigenvalue counting measure always exists…
We examine the adjacency matrices of three-regular graphs representing one-face maps. Numerical studies reveal that the limiting eigenvalue statistics of these matrices are the same as those of much larger, and more widely studied classes…
In this note we compare two recent results about the distribution of eigenvalues for semi-classical pseudodifferential operators in two dimensions. For classes of analytic operators A. Melin and the author obtained a complex Bohr-Sommerfeld…