Random Matrices: Theory and Applications

We consider the singular linear statistic of the Laguerre unitary ensemble (LUE) consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular $K$ Bessel functions. Earlier studies have identified the same determinant, but with the $K$ Bessel functions replaced by $I$ Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the LUE, in the case of non-negative integer Laguerre parameter. We show that the Toeplitz determinant formed from an arbitrary linear combination of these two Bessel functions occurs as a $\tau$-function sequence in Okamoto’s Hamiltonian formulation of Painlevé III${}^{\prime }$, and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same $\sigma$-form Painlevé III${}^{\prime }$ differential equation, giving an explanation of a fact which can be observed from earlier results. In addition, some insights into the relationship between this characterization of the generating function, and its characterization in the $n\to \infty$ limit, both with the Laguerre parameter $\alpha$ fixed, and with $\alpha =n$ (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.

We characterize the limiting fluctuations of traces of several independent Wigner matrices and deterministic matrices under mild conditions. A CLT holds but in general the families are not asymptotically free of second-order and the limiting covariance depends the limiting ${}^{\ast }$-distribution of the deterministic matrices and their transposes and Hadamard products.

Assuming a covariance structure with blocked compound symmetry, it was showed that unbiased estimators for the covariance matrices are optimal under normality. In this paper, we derive the asymptotic distribution of the correlation matrix using unbiased estimators and discuss its use in hypothesis testing. The accuracy of the result is investigated through numerical simulation and the method is applied to real data.

We establish a moderate deviation principle for linear eigenvalue statistics of $\beta$-ensembles in the one-cut regime with a real-analytic potential. The main ingredient is to obtain uniform estimates for the correlators of a family of perturbations of $\beta$-ensembles using the loop equations.

We prove a central limit theorem (CLT) for the product of a class of random singular matrices related to a random Hill’s equation studied by Adams–Bloch–Lagarias. The CLT features an explicit formula for the variance in terms of the distribution of the matrix entries and this allows for exact calculation in some examples. Our proof relies on a novel connection to the theory of m-dependent sequences which also leads to an interesting and precise nondegeneracy condition.

We find the asymptotic spectral distribution of random Kummer matrix. Then we formulate and prove a free analogue of HV independence property, which is known for classical Kummer and Gamma random variables and for Kummer and Wishart matrices. We also prove a related characterization of free-Kummer and free-Poisson (Marchenko–Pastur) non-commutative random variables.

The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well-known analogy with the Boltzmann factor for a classical log-gas with pair potential $-log|x-y|$, confined by a one-body harmonic potential. A generalization is to replace the pair potential by $-log|sinh(\pi (x-y)/L)|$. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time $t=0$, and subsequently in the study of quantum many-body systems of the Calogero–Sutherland type, and also in Chern–Simons field theory. It is an example of a determinantal point process with correlation kernel based on the Stieltjes–Wigert polynomials. We take up the problem of determining the moments of this ensemble, and find an exact expression in terms of a particular little q-Jacobi polynomial. From their large N form, the global density can be computed. Previous work has evaluated the edge scaling limit of the correlation kernel in terms of the Ramanujan ($q$-Airy) function. We show how in a particular $L\to \infty$ scaling limit, this reduces to the Airy kernel.

The scaled standard Wigner matrix (symmetric with mean zero, variance one i.i.d. entries), and its limiting eigenvalue distribution, namely the semi-circular distribution, have attracted much attention. The $2k$th moment of the limit equals the number of non-crossing pair-partitions of the set $\{1,2,\dots ,2k\}$. There are several extensions of this result in the literature. In this paper, we consider a unifying extension which also yields additional results. Suppose $W_n$ is an $n\times n$ symmetric matrix where the entries are independently distributed. We show that under suitable assumptions on the entries, the limiting spectral distribution exists in probability or almost surely. The moments of the limit can be described through a set of partitions which in general is larger than the set of non-crossing pair-partitions. This set gives rise to interesting enumerative combinatorial problems. Several existing limiting spectral distribution results follow from our results. These include results on the standard Wigner matrix, the adjacency matrix of a sparse homogeneous Erdős–Rényi graph, heavy tailed Wigner matrix, some banded Wigner matrices, and Wigner matrices with variance profile. Some new results on these models and their extensions also follow from our main results.

We consider the probability that no points lie on g large intervals in the bulk of the Airy point process. We make a conjecture for all the terms in the asymptotics up to and including the oscillations of order 1, and we prove this conjecture for g = 1.