Related papers: The Singular Values of the GUE (Less is More)
We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with a pole of order $k$ at the origin, in…
The focus of this survey paper is on the distribution function for the largest eigenvalue in the finite N Gaussian ensembles (GOE,GUE,GSE) in the edge scaling limit of N->infinity. These limiting distribution functions are expressible in…
Statistics over the Gaussian unitary ensemble and the Wishart ensemble of random matrices often have nice closed-form expressions. These are related to multivariate extensions of the Hermite, Laguerre, and Jacobi polynomials, which often…
We study the eigenvalue correlations of random Hermitian $n\times n$ matrices of the form $S=M+\epsilon H$, where $H$ is a GUE matrix, $\epsilon>0$, and $M$ is a positive-definite Hermitian random matrix, independent of $H$, whose…
We study eigenvectors in the deformed Gaussian unitary ensemble of random matrices $H=W\tilde{H}W$, where $\tilde{H}$ is a random matrix from Gaussian unitary ensemble and $W$ is a deterministic diagonal matrix with positive entries. Using…
We study the asymptotic behavior of the eigenvalues of Gaussian perturbations of large Hermitian random matrices for which the limiting eigenvalue density vanishes at a singular interior point or vanishes faster than a square root at a…
Akemann, Ipsen, and Kieburg showed recently that the squared singular values of a product of M complex Ginibre matrices are distributed according to a determinantal point process. We introduce the notion of a polynomial ensemble and show…
We prove that the squared singular values of a fixed matrix multiplied with a truncation of a Haar distributed unitary matrix are distributed by a polynomial ensemble. This result is applied to a multiplication of a truncated unitary matrix…
We introduce a new family of $N\times N$ random real symmetric matrix ensembles, the $k$-checkerboard matrices, whose limiting spectral measure has two components which can be determined explicitly. All but $k$ eigenvalues are in the bulk,…
Introducing sets of constraints, we define new classes of random-matrix ensembles, the constrained Gaussian unitary (CGUE) and the deformed Gaussian unitary (DGUE) ensembles. The latter interpolate between the GUE and the CGUE. We derive a…
Assume a finite set of complex random variables form a determinantal point process, we obtain a theorem on the limit of the empirical distribution of these random variables. The result is applied to %We study the limits of the empirical…
We derive simple linear, inhomogeneous recurrences for the variance of the index by utilising the fact that the generating function for the distribution of the number of positive eigenvalues of a Gaussian unitary ensemble is a…
We study the probability that all the eigenvalues of $n\times n$ Hermitian matrices, from the Laguerre unitary ensemble with the weight $x^{\gamma}\mathrm{e}^{-4nx},\;x\in[0,\infty),\;\gamma>-1$, lie in the interval $[0,\alpha]$. By using…
These notes provide an introduction to the theory of random matrices. The central quantity studied is $\tau(a)= det(1-K)$ where $K$ is the integral operator with kernel $1/\pi} {\sin\pi(x-y)\over x-y} \chi_I(y)$. Here…
Observables in random tensor theory are polynomials in the entries of a tensor of rank $d$ which are invariant under $U(N)^d$. It is notoriously difficult to evaluate the expectations of such polynomials, even in the Gaussian distribution.…
We are interested in two random matrix ensembles related to permutations: the ensemble of permutation matrices following Ewens' distribution of a given parameter $\theta >0$, and its modification where entries equal to $1$ in the matrices…
Recently Burkhardt et. al. introduced the $k$-checkerboard random matrix ensembles, which have a split limiting behavior of the eigenvalues (in the limit all but $k$ of the eigenvalues are on the order of $\sqrt{N}$ and converge to…
We study the eigenvalues and the eigenvectors of $N\times N$ structured random matrices of the form $H = W\tilde{H}W+D$ with diagonal matrices $D$ and $W$ and $\tilde{H}$ from the Gaussian Unitary Ensemble. Using the supersymmetry technique…
The squared singular values of the product of $M$ complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a hard edge scaling to a form specified in…
We consider powers of random matrices with independent entries. Let $X_{ij}, i,j\ge 1$, be independent complex random variables with $\E X_{ij}=0$ and $\E |X_{ij}|^2=1$ and let $\mathbf X$ denote an $n\times n$ matrix with $[\mathbf…