Invariant sums of random matrices and the onset of level repulsion

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) and yet the interaction between eigenvalues is not Vandermondian. The ensemble contains real symmetric or complex hermitian matrices $\mathbf{S}$ of the form $\mathbf{S}=\sum_{i=1}^M \langle \mathbf{O}_i \mathbf{D}_i\mathbf{O}_i^{\mathrm{T}}\rangle$ or $\mathbf{S}=\sum_{i=1}^M \langle \mathbf{U}_i \mathbf{D}_i\mathbf{U}_i^\dagger\rangle$ respectively. The diagonal matrices $\mathbf{D}_i=\mathrm{diag}\{\lambda_1^{(i)},\ldots,\lambda_N^{(i)}\}$ are constructed from real eigenvalues drawn \emph{independently} from distributions $p^{(i)}(x)$, while the matrices $\mathbf{O}_i$ and $\mathbf{U}_i$ are all orthogonal or unitary. The average $\langle\cdot\rangle$ is simultaneously performed over the symmetry group and the joint distribution of $\{\lambda_j^{(i)}\}$. We focus on the limits i.) $N\to\infty$ and ii.) $M\to\infty$, with $N=2$. In the limit i.), the resulting sum $\mathbf{S}$ develops level repulsion even though the original matrices do not feature it, and classical RMT universality is restored asymptotically. In the limit ii.) the spacing distribution attains scaling forms that are computed exactly: for the orthogonal case, we recover the $\beta=1$ Wigner's surmise, while for the unitary case an entirely new universal distribution is obtained. Our results allow to probe analytically the microscopic statistics of the sum of random matrices that become asymptotically free. We also give an interpretation of this model in terms of radial random walks in a matrix space. The analytical results are corroborated by numerical simulations.Comment: 19 pag., 6 fig. - published versio

Moments of random matrices and hypergeometric orthogonal polynomials

We establish a new connection between moments of $n \times n$ random matrices $X_n$ and hypergeometric orthogonal polynomials. Specifically, we consider moments \mathbb{E}\Tr X_n^{-s} as a function of the complex variable $s \in \mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. An application of the theory resolves part of an integrality conjecture of Cunden \textit{et al.}~[F. D. Cunden, F. Mezzadri, N. J. Simm and P. Vivo, J. Math. Phys. 57 (2016)] on the time-delay matrix of chaotic cavities. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order $n\to\infty$ asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials

A note on quantum chaology and gamma approximations to eigenvalue spacings for infinite random matrices

Quantum counterparts of certain simple classical systems can exhibit chaotic behaviour through the statistics of their energy levels and the irregular spectra of chaotic systems are modelled by eigenvalues of infinite random matrices. We use known bounds on the distribution function for eigenvalue spacings for the Gaussian orthogonal ensemble (GOE) of infinite random real symmetric matrices and show that gamma distributions, which have an important uniqueness property, can yield an approximation to the GOE distribution. That has the advantage that then both chaotic and non chaotic cases fit in the information geometric framework of the manifold of gamma distributions, which has been the subject of recent work on neighbourhoods of randomness for general stochastic systems. Additionally, gamma distributions give approximations, to eigenvalue spacings for the Gaussian unitary ensemble (GUE) of infinite random hermitian matrices and for the Gaussian symplectic ensemble (GSE) of infinite random hermitian matrices with real quaternionic elements, except near the origin. Gamma distributions do not precisely model the various analytic systems discussed here, but some features may be useful in studies of qualitative generic properties in applications to data from real systems which manifestly seem to exhibit behaviour reminiscent of near-random processes.Comment: 9 pages, 5 figures, 2 tables, 27 references. Updates version 1 with data and references from feedback receive