Random Matrices: Theory and Applications

For each n, let $U_n$ be Haar distributed on the group of $n\times n$ unitary matrices. Let $x_{n,1},\dots ,x_{n,m}$ denote orthogonal nonrandom unit vectors in ${ℂ}^n$ and let $u_{n,k}={(u_k^1,\dots ,u_k^n)}^{\ast }=U_n^{\ast }{x}_{n,k}$, $k=1,\dots ,m$. Define the following functions on $[0,1]$: $X_n^{k,k}(t)=\sqrt{n}{\sum }_{i=1}^{[nt]}(|u_k^i{|}^2-\frac{1}{n})$, $X_n^{k,k^{\prime }}(t)=\sqrt{2n}{\sum }_{i=1}^{[nt]}{ū}_k^i{u}_{k^{\prime }}^i$, $k<k^{\prime }$. Then it is proven that $X_n^{k,k},\Re X_n^{k,k^{\prime }}$, $\Im X_n^{k,k^{\prime }}$, considered as random processes in $D[0,1]$, converge weakly, as $n\to \infty$, to $m^2$ independent copies of Brownian bridge. The same result holds for the $m(m+1)/2$ processes in the real case, where $O_n$ is real orthogonal Haar distributed and $x_{n,i}\in {ℝ}^n$, with $\sqrt{n}$ in $X_n^{k,k}$ and $\sqrt{2n}$ in $X_n^{k,k^{\prime }}$ replaced with $\sqrt{\frac{n}{2}}$ and $\sqrt{n}$, respectively. This latter result will be shown to hold for the matrix of eigenvectors of $M_n=(1/s)V_n{V}_n^T$ where $V_n$ is $n\times s$ consisting of the entries of $\{v_{ij}\},i,j=1,2,\dots$, i.i.d. standardized and symmetrically distributed, with each $x_{n,i}=\{\pm 1/\sqrt{n},\dots ,\pm 1/\sqrt{n}\}$ and $n/s\to y>0$ as $n\to \infty$. This result extends the result in [J. W. Silverstein, Ann. Probab. 18 (1990) 1174–1194]. These results are applied to the detection problem in sampling random vectors mostly made of noise and detecting whether the sample includes a nonrandom vector. The matrix $B_n=𝜃v_n{v}_n^{\ast }+S_n$ is studied where $S_n$ is Hermitian or symmetric and nonnegative definite with either its matrix of eigenvectors being Haar distributed, or $S_n=M_n$, $𝜃>0$ nonrandom and $v_n$ is a nonrandom unit vector. Results are derived on the distributional behavior of the inner product of vectors orthogonal to $v_n$ with the eigenvector associated with the largest eigenvalue of $B_n.$

In this paper, we derive the exact distributions of eigenvalues of a singular Wishart matrix under the elliptical model. We define the generalized heterogeneous hypergeometric functions with two matrix arguments and provide the convergence conditions of these functions. The joint density of eigenvalues and the distribution function of the largest eigenvalue for a singular elliptical Wishart matrix are represented with these functions. Numerical computations for the distribution of the largest eigenvalue are conducted under the matrix-variate t and Kotz type models.

The loop equations for the $\beta$-ensembles are conventionally solved in terms of a $1/N$ expansion. We observe that it is also possible to fix N and expand in inverse powers of $\beta$. At leading order, for the one-point function $W_1(x)$ corresponding to the average of the linear statistic $A={\sum }_{j=1}^{N}1/(x-{\lambda }_j)$ and after specialising to the classical weights, this reclaims well known results of Stieltjes relating the zeros of the classical polynomials to the minimum energy configuration of certain log–gas potential energies. Moreover, it is observed that the differential equations satisfied by $W_1(x)$ in the case of classical weights — which are particular Riccati equations — are simply related to the differential equations satisfied by $W_1(x)$ in the high temperature scaled limit $\beta =2\alpha /N$ ($\alpha$ fixed, $N\to \infty$), implying a certain high–low temperature duality. A generalisation of this duality, valid without any limiting procedure, is shown to hold for $W_1(x)$ and all its higher point analogues in the classical $\beta$-ensembles.

We consider a correlated $N\times N$ Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one.

Maps are polygonal cellular networks on Riemann surfaces. This paper analyzes the construction of closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. The method of construction developed here involves a novel asymptotic symbol calculus for difference operators based on the relation between spectral asymptotics for Hermitian random matrices and asymptotics of orthogonal polynomials with exponential weights. These closed form expressions have a universal character in the sense that they are independent of the explicit valence distribution of the cellular networks within a broad class. Nevertheless the valence distributions may be recovered from the closed form generating functions by a remarkable unwinding identity in terms of Appell polynomials generated by Bessel functions. Our treatment reveals the generating functions to be solutions of nonlinear conservation laws and their prolongations. This characterization enables one to gain insights that go beyond more traditional methods that are purely combinatorial. Universality results are connected to stability results for characteristic singularities of conservation laws that were studied by Caflisch, Ercolani, Hou and Landis, Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems, Commun. Pure Appl. Math. 46 (1993) 453–499, as well as directly related to universality results for random matrix spectra.

Recently, tensors are widely used to represent higher-order data with internal spatial or temporal relations, e.g. images, videos, hyperspectral images (HSIs). While the true signals are usually corrupted by noises, it is of interest to study tensor recovery problems. To this end, many models have been established based on tensor decompositions. Traditional tensor decomposition models, such as the CP and Tucker factorization, treat every mode of tensors equally. However, in many real applications, some modes of the data act differently from the other modes, e.g. channel mode of images, time mode of videos, band mode of HSIs. The recently proposed model called t-SVD aims to tackle such problems. In this paper, we focus on tensor denoising problems. Specifically, in order to obtain low-rank estimators of true signals, we propose to use different shrinkage functions to shrink the tensor singular values based on the t-SVD. We derive Stein’s unbiased risk estimate (SURE) of the proposed model and develop adaptive SURE-based tuning parameter selection procedure, which is totally data-driven and simultaneous with the estimation process. The whole modeling procedure requires only one round of t-SVD. To demonstrate our model, we conduct experiments on simulation data, images, videos and HSIs. The results show that the proposed SURE approximates the true risk function accurately. Moreover, the proposed model selection procedure picks good tuning parameters out. We show the superiority of our model by comparing with state-of-the-art denoising models. The experiments manifest that our model outperforms in both quantitative metrics (e.g. RSE, PSNR) and visualizing results.

We prove large deviations principles for spectral measures of perturbed (or spiked) matrix models in the direction of an eigenvector of the perturbation. In each model under study, we provide two approaches, one of which relying on large deviations principle of unperturbed models derived in the previous work “Sum rules via large deviations” (Gamboa et al. [Sum rules via large deviations, J. Funct. Anal. 270(2) (2016) 509–559]).

A famous result going back to Eric Kostlan states that the moduli of the eigenvalues of random normal matrices with radial potential are independent yet non-identically distributed. This phenomenon is at the heart of the asymptotic analysis of the edge, and leads in particular to the Gumbel fluctuation of the spectral radius when the potential is quadratic. In the present work, we show that a wide variety of laws of fluctuation are possible, beyond the already known cases, including for instance Gumbel and exponential laws at unusual speeds. We study the convergence in law of the spectral radius as well as the limiting point process at the edge. Our work can also be seen as the asymptotic analysis of the edge of two-dimensional determinantal Coulomb gases and the identification of the limiting kernels.

We use techniques in the shuffle algebra to present a formula for the partition function of a one-dimensional log-gas comprised of particles of (possibly) different integer charges at certain inverse temperature $\beta$ in terms of the Berezin integral of an associated non-homogeneous alternating tensor. This generalizes previously known results by removing the restriction on the number of species of odd charge. Our methods provide a unified framework extending the de Bruijn integral identities from classical $\beta$-ensembles ($\beta =1,2,4$) to multicomponent ensembles, as well as to iterated integrals of more general determinantal integrands.

Poisson thinning is an elementary result in probability, which is of great importance in the theory of Poisson point processes. In this paper, we record a couple of characterization results on Poisson thinning. We also consider several free probability analogues of Poisson thinning, which we collectively dub as free Poisson, and prove characterization results for them, similar to the classical case. One of these free Poisson thinning procedures arises naturally as a high-dimensional asymptotic analogue of Cochran’s theorem from multivariate statistics on the “Wishart-ness” of quadratic functions of Gaussian random matrices. We note the implications of our characterization results in the context of Cochran’s theorem. We also prove a free probability analogue of Craig’s theorem, another well-known result in multivariate statistics on the independence of quadratic functions of Gaussian random matrices.

We prove an operator level limit for the circular Jacobi $\beta$-ensemble. As a result, we characterize the counting function of the limit point process via coupled systems of stochastic differential equations. We also show that the normalized characteristic polynomials converge to a random analytic function, which we characterize via the joint distribution of its Taylor coefficients at zero and as the solution of a stochastic differential equation system. We also provide analogous results for the real orthogonal $\beta$-ensemble.