Products of shifted characteristic polynomials, and ratios of such products, averaged over the classical compact groups are of great interest to number theorists as they model similar averages of L-functions in families with the same symmetry type as the compact group. We use Toeplitz and Toeplitz plus Hankel operators and the identities of Borodin–Okounkov–Case–Geronimo, and Basor–Ehrhardt to prove

We provide a dynamical study of a model of multiplicative perturbation of a unitary matrix introduced by Fyodorov. In particular, we identify a flow of deterministic domains that bound the spectrum with high probability, separating the outlier from the typical eigenvalues at all sub-critical timescales. These results are obtained under generic assumptions on U that hold for a variety of unitary random

It is well known that the semi-circle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko–Pastur theorems. In this work, we shed the light on an intriguing phenomenon suggesting that this functional is monotonic along the mean empirical spectral distribution in

In this paper, we study high-dimensional behavior of empirical spectral distributions {LN(t),t∈[0,T]} for a class of N×N symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic differential equation driven by fractional Brownian motion with Hurst parameter H∈(1/2,1). For Wigner-type matrices, we obtain almost sure relative compactness of {LN(t),t∈[0,T]}N∈ℕ in

Starting from Montgomery’s conjecture, there has been a substantial interest on the connections of random matrix theory and the theory of L-functions. In particular, moments of characteristic polynomials of random matrices have been considered in various works to estimate the asymptotics of moments of L-function families. In this paper, we first consider joint moments of the characteristic polynomial

In this paper, we consider the elliptic Ginibre matrices in the orthogonal symmetry class that interpolates between the real Ginibre ensemble and the Gaussian orthogonal ensemble. We obtain the finite size corrections of the real eigenvalue densities in both the global and edge scaling regimes, as well as in both the strong and weak non-Hermiticity regimes. Our results extend and provide the rate of

Voiculescu’s freeness emerges when computing the asymptotic spectra of polynomials on N×N random matrices with eigenspaces in generic positions: they are randomly rotated with a uniform unitary random matrix UN. In this paper, we elaborate on the previous result by proposing a random matrix model, which we name the Vortex model, where UN has the law of a uniform unitary random matrix conditioned to

We present a simple stochastic representation for the joint distribution of sample estimates of three scalar parameters and two vectors of portfolio weights that characterize the minimum-variance frontier. This stochastic representation is useful for sampling observations efficiently, deriving moments in closed-form, and studying the distribution and performance of many portfolio strategies that are

In the paper, we consider the extended Gross–Witten–Wadia unitary matrix model by introducing a logarithmic term in the potential. The partition function of the model can be expressed equivalently in terms of the Toeplitz determinant with the (i,j)-entry being the modified Bessel functions of order i−j−ν, ν∈ℂ. When the degree n is finite, we show that the Toeplitz determinant is described by the isomonodromy

In [W. Ejsmont and F. Lehner, The free tangent law, Adv. Appl. Math. 121 (2020) 102093], we study the limit sums of free commutators and anticommutators and show that the generalized tangent function tanz1−xtanz describes the limit distribution. This is the generating function of the higher order tangent numbers of Carlitz and Scoville (see (1.6) in [L. Carlitz and R. Scoville, Tangent numbers and

In this paper, we study a nonlinear second-order ordinary differential equation which we call the Ermakov–Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced either to the Ermakov or the Painlevé XXV equation. The Ermakov–Painlevé XXV equation arises from a generalized Riccati equation and the related third-order linear differential equation via the Schwarzian

Let X be an n×n symmetric random matrix with independent but non-identically distributed entries. The deviation inequalities of the spectral norm of X with Gaussian entries have been obtained by using the standard concentration of Gaussian measure results. This paper establishes an upper tail bound of the spectral norm of X with sub-exponential entries. Our method relies upon a crucial ingredient of

A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank 1 perturbation. Considered in this review are the additive rank 1 perturbation of the Hermitian Gaussian ensembles, the multiplicative rank 1 perturbation of the Wishart ensembles, and rank 1 perturbations of Hermitian and unitary matrices giving rise

The stochastic block model (SBM) is an extension of the Erdős–Rényi graph and has applications in numerous fields, such as data analysis, recovering community structure in graph data and social networks. In this paper, we consider the normal central SBM adjacency matrix with K communities of arbitrary sizes. We derive an explicit formula for the limiting empirical spectral density function when the

Motivated by the general matrix deviation inequality for i.i.d. ensemble Gaussian matrix [R. Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, 2018), doi:10.1017/9781108231596 of Theorem 11.1.5], we show that this property holds for the ℓp-norm with 1≤p<∞ and i.i.d. ensemble

Motivated by the stochastic block model, we investigate a class of Wigner-type matrices with certain block structures and establish a CLT for the corresponding linear spectral statistics (LSS) via the large-deviation bounds from local law and the cumulant expansion formula. We apply the results to the stochastic block model. Specifically, a class of renormalized adjacency matrices will be block-Wigner-type

In the regime where the parameter beta is proportional to the reciprocal of the system size, it is known that the empirical distribution of Gaussian beta ensembles (respectively, beta Laguerre ensembles) converges weakly to a probability measure of associated Hermite polynomials (respectively, associated Laguerre polynomials), almost surely. Gaussian fluctuations around the limit have been known as

We compute the eigenvalue fluctuations of uniformly distributed random biregular bipartite graphs with fixed and growing degrees for a large class of analytic functions. As a key step in the proof, we obtain a total variation distance bound for the Poisson approximation of the number of cycles and cyclically non-backtracking walks in random biregular bipartite graphs, which might be of independent

Let x1,…,xn be independent and identically distributed (i.i.d.) real-valued random vectors from distribution Np(μ,Σ), where the sample size n and the vector dimension p satisfy n−1>p→∞. We are interested in the exponential convergence rate of the likelihood ratio test (LRT) statistics for testing Σ equal to a given matrix and (μ,Σ) equal to a given pair. In traditional statistical theory, the LRT statistics

In this paper, we give an analytic proof of the asymptotic behavior of the moments of moments of the characteristic polynomials of random symplectic and orthogonal matrices. We therefore obtain alternate, integral expressions for the leading order coefficients previously found by Assiotis, Bailey and Keating. We also discuss the conjectures of Bailey and Keating for the corresponding moments of moments

In this paper, for the generalized estimating equation (GEE) with diverging number of covariates, the asymptotic properties of GEE estimator are considered. Under the weaker assumption on the minimum eigenvalue of Fisher information matrix and some other regular conditions, we prove the asymptotic existence, consistency and asymptotic normality of the GEE estimator and the asymptotic distribution of

The density matrix formalism is a fundamental tool in studying various problems in quantum information processing. In the space of density matrices, the most well-known measures are the Hilbert–Schmidt and Bures–Hall ensembles. In this work, the averages of quantum purity and von Neumann entropy for an ensemble that interpolates between these two major ensembles are explicitly calculated for finite-dimensional

In this paper, we are concerned with nonlinear interaction detection based on partial dimension reduction with missing response data. The covariates are grouped through linear combinations in a general class of semi-parametric models to detect their joint interaction effects. The joint interaction effects are estimated by a profile least squares approach with the help of the inverse probability weighted

In this paper, we analyze the covariance kernel of the Gaussian process that arises as the limit of fluctuations of linear spectral statistics for Wigner matrices with a few moments. More precisely, the process we study here corresponds to Hermitian matrices with independent entries that have α moments for 2<α<4. We obtain a closed form α-dependent expression for the covariance of the limiting process

Consider the empirical autocovariance matrices at given non-zero time lags, based on observations from a multivariate complex Gaussian stationary time series. The spectral analysis of these autocovariance matrices can be useful in certain statistical problems, such as those related to testing for white noise. We study the behavior of their spectral measure in the asymptotic regime where the time series

In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In [6], the first and second authors provided conditions under which tensor products of unitary random matrices are asymptotically free with respect to the normalized trace. Here, we extend this result by proving that asymptotic freeness of tensor products of Haar unitary matrices

We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups O(N), U(N) and Sp(2N). For finite matrix size we calculate the moments in terms of hypergeometric functions of matrix argument and give explicit integral representations highlighting the duality between the moment and the matrix size as well as the duality between the orthogonal

This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices 1N(Dn∘Xn)(Dn∘Xn)∗, studied in [V. L. Girko, Theory of Stochastic Canonical Equations: Vol. 1 (Kluwer Academic Publishers, Dordrecht, 2001)]. Here, Xn=(xij) is an n×N random matrix consisting of independent complex standardized random variables, Dn=(dij), n×N, has nonnegative entries, and ∘ denotes Hadamard

In this paper, we propose a new approach to the central limit theorem (CLT) based on functions of bounded Fréchet variation for the continuously differentiable linear statistics of random matrix ensembles which relies on a weaker form of a large deviation principle for the operator norm; a Poincaré-type inequality for the linear statistics; and the existence of a second-order limit distribution. This

We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the convergence rates of the correlation functions at the real bulk/edge of the spectrum, which in particular establishes the local universality at strong non-Hermiticity

In [I. Dumitriu and E. Paquette, Spectra of overlapping Wishart matrices and the gaussian free field, Random Matrices: Theory Appl. 07(2) (2018) 1850003], the authors consider eigenvalues of overlapping Wishart matrices and prove that its fluctuations asymptotically convergence to the Gaussian free field. In this brief note, their result is extended to show that when the matrix entries undergo stochastic

We consider random Hermitian matrices with independent upper triangular entries. Wigner’s semicircle law says that under certain additional assumptions, the empirical spectral distribution converges to the semicircle distribution. We characterize convergence to semicircle in terms of the variances of the entries, under natural assumptions such as the Lindeberg condition. The result extends to certain

We study the eigenvalue distribution of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the analog of the population covariance matrix is now a function of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated

We prove an operator level limit for the circular Jacobi β-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

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

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 β 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

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

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.

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

For each n, let Un be Haar distributed on the group of n×n unitary matrices. Let xn,1,…,xn,m denote orthogonal nonrandom unit vectors in ℂn and let un,k=(uk1,…,ukn)∗=Un∗xn,k, k=1,…,m. Define the following functions on [0,1]: Xnk,k(t)=n∑i=1[nt](|uki|2−1n), Xnk,k′(t)=2n∑i=1[nt]ūkiuk′i, k0 as n→∞. This result extends the result in [J. W. Silverstein, Ann. Probab. 18 (1990) 1174–1194]. These results are

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

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

In this paper, a new concept for a 3 × 3 block relation matrix is studied in a Banach space. It is shown that, under certain condition, we can investigate the Frobenius–Schur decomposition of relation matrices. Furthermore, we present some conditions which should allow the multivalued 3 × 3 matrices linear operator to be closable.

The loop equations for the β-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 β. At leading order, for the one-point function W1(x) corresponding to the average of the linear statistic A=∑j=1N1/(x−λj) and after specialising to the classical weights, this reclaims well known results of Stieltjes relating the

The loop equations for the β-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 β. At leading order, for the one-point function W1(x) corresponding to the average of the linear statistic A=∑j=1N1/(x−λj) and after specialising to the classical weights, this reclaims well known results of Stieltjes relating the

We consider a correlated N×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.

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

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 2kth moment of the limit equals the number of non-crossing pair-partitions of the set {1, 2,…, 2k}. There are several extensions of this result in the literature. In this paper, we consider a unifying

For N,n ∈ ℕ, consider the sample covariance matrix SN(T) = 1 NXX∗ from a data set X = CN1/2ZT n1/2, where Z = (Zi,j) is a N × n matrix having i.i.d. entries with mean zero and variance one, and CN,Tn are deterministic positive semi-definite Hermitian matrices of dimension N and n, respectively. We assume that (CN)N is bounded in spectral norm, and Tn is a Toeplitz matrix with its largest eigenvalues

The aim of this paper is to investigate the spectral properties of sample covariance matrices under a more general population. We consider a class of matrices of the form Sn = 1 nBnXnXn∗B n∗, where Bn is a p × m nonrandom matrix and Xn is an m × n matrix consisting of i.i.d standard complex entries. p/n → c ∈ (0,∞) as n →∞ while m can be arbitrary but no smaller than p. We first prove that under some

We extend recent results on the asymptotic eigenvalue distribution of the SYK model to the multivariate case and relate the limit of a dynamical version of the SYK model with the q-Brownian motion, a non-commutative deformation of classical Brownian motion. Furthermore, we extend the results for fluctuations to the multivariate setting and treat also higher correlation functions. The structure of our

We extend recent results on the asymptotic eigenvalue distribution of the SYK model to the multivariate case and relate the limit of a dynamical version of the SYK model with the q-Brownian motion, a non-commutative deformation of classical Brownian motion. Furthermore, we extend the results for fluctuations to the multivariate setting and treat also higher correlation functions. The structure of our

Given n,m ∈ ℕ, we study two classes of large random matrices of the form ℒn =∑α=1mξ αyαyαTand𝒜 n =∑α=1mξ α(yαxαT + x αyαT), where for every n, (ξα)α are iid copies of a random variable ξ = ξ(n) ∈ ℝ, (xα)α, (yα)α ⊂ ℝn are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic

We consider linear spectral statistics built from the block-normalized correlation matrix of a set of M mutually independent scalar time series. This matrix is composed of M2 blocks. Each block has size L×L and contains the sample cross-correlation measured at L consecutive time lags between each pair of time series. Let N denote the total number of consecutively observed windows that are used to estimate

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures Ξ on a space S with measure λ, whose correlation functions are all given by determinants specified by an integral kernel K called the correlation kernel. We consider a pair of Hilbert spaces, Hℓ,ℓ=1,2, which are assumed to be realized as L2-spaces, L2(Sℓ,λℓ), ℓ=1,2, and introduce a bounded linear

In this paper, we study the condition number of a random Toeplitz matrix. As a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategies to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding as a decoupling

We consider linear spectral statistics built from the block-normalized correlation matrix of a set of M mutually independent scalar time series. This matrix is composed of M2 blocks. Each block has size L × L and contains the sample cross-correlation measured at L consecutive time lags between each pair of time series. Let N denote the total number of consecutively observed windows that are used to

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures Ξ on a space S with measure λ, whose correlation functions are all given by determinants specified by an integral kernel K called the correlation kernel. We consider a pair of Hilbert spaces, Hℓ,ℓ = 1, 2, which are assumed to be realized as L2-spaces, L2(S ℓ,λℓ), ℓ = 1, 2, and introduce a bounded

In this paper, we study the condition number of a random Toeplitz matrix. As a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategies to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding as a decoupling

In this paper, we consider the spectrum of a Laplacian matrix, also known as Markov matrices where the entries of the matrix are independent but have a variance profile. Motivated by recent works on generalized Wigner matrices we assume that the variance profile gives rise to a sequence of graphons. Under the assumption that these graphons converge, we show that the limiting spectral distribution converges