Random Matrices: Theory and Applications

Consider the ensembles of real symmetric Toeplitz matrices and real symmetric Hankel matrices whose entries are i.i.d. random variables chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous work on real symmetric Toeplitz matrices shows that the spectral measures, or densities of normalized eigenvalues, converge almost surely to a universal near-Gaussian distribution, while previous work on real symmetric Hankel matrices shows that the spectral measures converge almost surely to a universal non-unimodal distribution. Real symmetric Toeplitz matrices are constant along the diagonals, while real symmetric Hankel matrices are constant along the skew diagonals. We generalize the Toeplitz and Hankel matrices to study matrices that are constant along some curve described by a real-valued bivariate polynomial (other authors refer to the dependencies among the matrix elements as arising from a link function). Using the Method of Moments and an analysis of the resulting Diophantine equations, we show that the spectral measures associated with linear bivariate polynomials converge in probability and almost surely to universal non-semicircular distributions. We prove that for certain choices these limiting distributions approach the semicircle in the limit of large values of the polynomial coefficients. We then prove that the spectral measures associated with the sum or difference of any two real-valued polynomials with different degrees converge in probability and almost surely to a universal semicircular distribution.

We investigate the universality of singular value and eigenvalue distributions of matrix-valued functions of independent random matrices and apply these general results in several examples. In particular we determine the limit distribution and prove universality under general conditions for singular value and eigenvalue distributions of products of independent matrices from spherical ensembles.

Let X be a random matrix whose pairs of entries X_jk and X_kj are correlated and vectors (X_jk, X_kj), for 1 ≤ j < k ≤ n, are mutually independent. Assume that the diagonal entries are independent from off-diagonal entries as well. We assume that , for any j, k = 1, …, n and 𝔼 X_jkX_kj = ρ for 1 ≤ j < k ≤ n. Let M_n be a non-random n × n matrix with ‖M_n‖ ≤ Kn^Q, for some positive constants K > 0 and Q ≥ 0. Let s_n(X + M_n) denote the least singular value of the matrix X + M_n. It is shown that there exist positive constants A and B depending on K, Q, ρ only such that

We study a new Selberg-type integral with n + m indeterminates, which turns out to be related to the deformed Calogero–Sutherland systems. We show that the integral satisfies a holonomic system of n + m non-symmetric linear partial differential equations. We also prove that a particular hypergeometric function defined in terms of super-Jack polynomials is the unique solution of the system. Some properties such as duality relations, integral formulas, Pfaff–Euler and Kummer transformations are also established. As a direct application, we evaluate the expectation value of ratios of characteristic polynomials in the classical β-ensembles of Random Matrix Theory.

A system of non-intersecting squared Bessel processes is considered which all start from one point and they all return to another point. Under the scaling of the starting and ending points when the macroscopic boundary of the paths touches the hard edge, a limiting critical process is described in the neighborhood of the touching point which we call the hard edge tacnode process. We derive its correlation kernel in an explicit new form which involves Airy type functions and operators that act on the direct sum of L²(ℝ₊) and a finite-dimensional space. As the starting points of the squared Bessel paths are set to 0, a cusp in the boundary appears. The limiting process is described near the cusp and it is called the hard edge Pearcey process. We compute its multi-time correlation kernel which extends the existing formulas for the single-time kernel. Our pre-asymptotic correlation kernel involves the ratio of two Toeplitz determinants which are rewritten using a Borodin–Okounkov type formula.

As a unifying framework for examining several properties that nominally involve eigenvalues, we present a particular structure of the singular values of the Gaussian orthogonal ensemble (GOE): the even-location singular values are distributed as the positive eigenvalues of a Gaussian ensemble with chiral unitary symmetry, while the odd-location singular values, conditioned on the even-location ones, can be algebraically transformed into a set of independent χ-distributed random variables. We discuss three applications of this structure: first, there is a pair of bidiagonal square matrices, whose singular values are jointly distributed as the even- and odd-location ones of the GOE; second, the magnitude of the determinant of the GOE is distributed as a product of simple independent random variables; third, on symmetric intervals, the gap probabilities of the GOE can be expressed in terms of the Laguerre unitary ensemble. We work specifically with matrices of finite order, but by passing to a large matrix limit, we also obtain new insight into asymptotic properties such as the central limit theorem of the determinant or the gap probabilities in the bulk-scaling limit. The analysis in this paper avoids much of the technical machinery (e.g. Pfaffians, skew-orthogonal polynomials, martingales, Meijer G-function, etc.) that was previously used to analyze some of the applications.