Related papers: Universality for random matrices with an edge spec…
Unitary random matrix ensembles Z_{n,N}^{-1} (\det M)^alpha exp(-N Tr V(M)) dM defined on positive definite matrices M, where alpha > -1 and V is real analytic, have a hard edge at 0. The equilibrium measure associated with V typically…
We prove universality at the edge of the spectrum for unitary (beta=2), orthogonal (beta=1) and symplectic (beta=4) ensembles of random matrices in the scaling limit for a class of weights w(x)=exp(-V(x)) where V is a polynomial,…
We consider symmetric and Hermitian random matrices whose entries are independent and symmetric random variables with an arbitrary variance pattern. Under a novel Short-to-Long Mixing condition, which is sharp in the sense that it precludes…
We give a proof of the Universality Conjecture for orthogonal (beta=1) and symplectic (beta=4) random matrix ensembles of Laguerre-type in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated…
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…
We give a proof of the Universality Conjecture for orthogonal and symplectic ensembles of random matrices in the scaling limit for a class of weights w(x)=exp(-V(x)) where V is a polynomial, V(x)=kappa_{2m}x^{2m}+..., kappa_{2m}>0. For such…
Universality of eigenvalue spacings is one of the basic characteristics of random matrices. We give the precise meaning of universality and discuss the standard universality classes (sine, Airy, Bessel) and their appearance in unitary,…
We compute analytically the joint probability density of eigenvalues and the level spacing statistics for an ensemble of random matrices with interesting features. It is invariant under the standard symmetry groups (orthogonal and unitary)…
We prove edge universality of local eigenvalue statistics for orthogonal invariant matrix models with real analytic potentials and one interval limiting spectrum. Our starting point is the result of \cite{S:08} on the representation of the…
Given an $n \times n$ complex matrix $A$, let $$\mu_{A}(x,y):= \frac{1}{n} |\{1\le i \le n, \Re \lambda_i \le x, \Im \lambda_i \le y\}|$$ be the empirical spectral distribution (ESD) of its eigenvalues $\lambda_i \in \BBC, i=1, ... n$. We…
It has been shown by Strahov and Fyodorov that averages of products and ratios of characteristic polynomials corresponding to Hermitian matrices of a unitary ensemble, involve kernels related to orthogonal polynomials and their Cauchy…
It is a result of Ginibre that the normalized bulk $k$-point correlation functions of a complex $n\times n$ Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process…
We establish universality of local eigenvalue correlations in unitary random matrix ensembles (1/Z_n) |\det M|^{2\alpha} e^{-n\tr V(M)} dM near the origin of the spectrum. If V is even, and if the recurrence coefficients of the orthogonal…
In this paper the kernel for the spectral correlation functions of the invariant chiral random matrix ensembles with real ($\beta =1$) and quaternion real ($\beta = 4$) matrix elements is expressed in terms of the kernel of the…
We consider unitary random matrix ensembles Z_{n,s,t}^{-1}e^{-n tr V_{s,t}(M)}dM on the space of Hermitian n x n matrices M, where the confining potential V_{s,t} is such that the limiting mean density of eigenvalues (as n\to\infty and…
We consider certain large random matrices, called random inner-product kernel matrices, which are essentially given by a nonlinear function $f$ applied entrywise to a sample-covariance matrix, $f(X^TX)$, where $X \in \mathbb{R}^{d \times…
A feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices. Study of such ensembles realised by matrices with Gaussian entries leads to statistical quantities…
We consider large non-Hermitian real or complex random matrices $X$ with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of…
Consider $N\times N$ Hermitian or symmetric random matrices $H$ where the distribution of the $(i,j)$ matrix element is given by a probability measure $\nu_{ij}$ with a subexponential decay. Let $\sigma_{ij}^2$ be the variance for the…
We investigate traces of powers of random matrices whose distributions are invariant under rotations (with respect to the Hilbert--Schmidt inner product) within a real-linear subspace of the space of $n\times n$ matrices. The matrices we…