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The spherical orthogonal, unitary, and symplectic ensembles (SOE/SUE/SSE) $S_\beta(N,r)$ consist of $N \times N$ real symmetric, complex hermitian, and quaternionic self-adjoint matrices of Frobenius norm $r$, made into a probability space…

We consider random matrix ensembles on the set of Hermitian matrices that are heavy tailed, in particular not all moments exist, and that are invariant under the conjugate action of the unitary group. The latter property entails that the…

Consider two types of products of independent random matrices, including products of Ginibre matrices and inverse Ginibre matrices and products of truncated Haar unitary matrices and inverse truncated Haar matrices. Each product matrix has…

We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant…

We show that some of the best-known matrix decompositions of some of the best-known random matrix ensembles give us the unique $G$-invariant uniform distributions on some of the best-known manifolds. The eigenvectors distributions of the…

Let H be a Haar distributed random matrix on the group of pxp real orthogonal matrices. Partition H into four blocks: (1) the (1,1) element, (2)the rest of the first row, (3) the rest of the first column, and (4)the remaining (p-1)x(p-1)…

Let $A$ and $B$ be two $N$ by $N$ deterministic Hermitian matrices and let $U$ be an $N$ by $N$ Haar distributed unitary matrix. It is well known that the spectral distribution of the sum $H=A+UBU^*$ converges weakly to the free additive…

Let $U$ and $V$ be two independent $N$ by $N$ random matrices that are distributed according to Haar measure on $U(N)$. Let $\Sigma$ be a non-negative deterministic $N$ by $N$ matrix. The single ring theorem [26] asserts that the empirical…

Let $U_n$ be an $n \times n$ Haar unitary matrix. In this paper, the asymptotic normality and independence of $\Tr U_n, \Tr U_n^2, ..., \Tr U_n^k$ are shown by using elementary methods. More generally, it is shown that the renormalized…

An ensemble of 2 x 2 pseudo-Hermitian random matrices is constructed that possesses real eigenvalues with level-spacing distribution exactly as for the Gaussian Unitary Ensemble found by Wigner. By a re-interpretation of Connes' spectral…

Let B be the largest spacing between adjacent eigenvalues of the Polyakov loop. We propose to employ the distribution of B as an order parameter for the finite temperature phase transition in SU(N) lattice gauge theories. Using smeared…

We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of…

We analyze properties of non-hermitian matrices of size M constructed as square submatrices of unitary (orthogonal) random matrices of size N>M, distributed according to the Haar measure. In this way we define ensembles of random matrices…

Using Grassmann variables and an analogy with two dimensional electrostatics, we obtain the average eigenvalue distribution $\rho(\omega)$ of ensembles of $N \times N$ asymmetrically diluted Hopfield matrices in the limit $N \rightarrow…

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite rank random matrices is found in terms of a Painlev\'e V system, and the solution of its associated linear isomonodromic system. In…

This paper centers on the limit eigenvalue distribution for random Vandermonde matrices with unit magnitude complex entries. The phases of the entries are chosen independently and identically distributed from the interval $[-\pi,\pi]$.…

In the current work, we study the eigenvalue distribution results of a class of non-normal matrix-sequences which may be viewed as a low rank perturbation, depending on a parameter $\beta>1$, of the basic Toeplitz matrix-sequence…

We study Haar unitary random matrices with permuted entries. For a sequence of permutations $\left(\sigma_N\right)_N$, where $\sigma_N$ acts on $N\times N$ matrices we identify conditions under which the $\ast$--distribution of permuted…

The Ginibre ensemble of complex random Hamiltonian matrices $H$ is considered. Each quantum system described by $H$ is a dissipative system and the eigenenergies $Z_{i}$ of the Hamiltonian are complex-valued random variables. For generic…

The density of complex eigenvalues of random asymmetric $N\times N$ matrices is found in the large-$N$ limit. The matrices are of the form $H_0+A$ where $A$ is a matrix of $N^2$ independent, identically distributed random variables with…