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Rank 1 perturbations in random matrix theory — A review of exact results Random Matrices Theory Appl. (IF 0.9) Pub Date : 20230819
Peter J. ForresterA 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

Limiting spectral distribution of stochastic block model Random Matrices Theory Appl. (IF 0.9) Pub Date : 20230718
Giap Van Su, MayRu Chen, MeiHui Guo, HaoWei HuangThe 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

Matrix deviation inequality for ℓpnorm Random Matrices Theory Appl. (IF 0.9) Pub Date : 20230615
YuanChung Sheu, TeChun WangMotivated by the general matrix deviation inequality for i.i.d. ensemble Gaussian matrix [R. Vershynin, HighDimensional 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 ℓpnorm with 1≤p<∞ and i.i.d. ensemble

Central limit theorem for linear spectral statistics of blockWignertype matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 20230607
Zhenggang Wang, Jianfeng YaoMotivated by the stochastic block model, we investigate a class of Wignertype matrices with certain block structures and establish a CLT for the corresponding linear spectral statistics (LSS) via the largedeviation 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 blockWignertype

Limit theorems for moment processes of beta Dyson’s Brownian motions and beta Laguerre processes Random Matrices Theory Appl. (IF 0.9) Pub Date : 20230314
Fumihiko Nakano, Hoang Dung Trinh, Khanh Duy TrinhIn 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

Global eigenvalue fluctuations of random biregular bipartite graphs Random Matrices Theory Appl. (IF 0.9) Pub Date : 20230218
Ioana Dumitriu, Yizhe ZhuWe 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 nonbacktracking walks in random biregular bipartite graphs, which might be of independent

The moderate deviation principles of likelihood ratio tests under alternative hypothesis Random Matrices Theory Appl. (IF 0.9) Pub Date : 20230109
Yansong Bai, Yong ZhangLet x1,…,xn be independent and identically distributed (i.i.d.) realvalued 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

Random matrix theory and moments of moments of Lfunctions Random Matrices Theory Appl. (IF 0.9) Pub Date : 20221208
J. C. Andrade, C. G. BestIn 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

Asymptotic properties of GEE with diverging dimension of covariates Random Matrices Theory Appl. (IF 0.9) Pub Date : 20221112
Chunhua Zhu, Qibing Gao, Yi YaoIn 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

Quantum interpolating ensemble: Biorthogonal polynomials and average entropies Random Matrices Theory Appl. (IF 0.9) Pub Date : 20221026
Lu Wei, Nicholas WitteThe density matrix formalism is a fundamental tool in studying various problems in quantum information processing. In the space of density matrices, the most wellknown 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 finitedimensional

Nonlinear interaction detection through partial dimension reduction with missing response data Random Matrices Theory Appl. (IF 0.9) Pub Date : 20221013
HongXia Xu, GuoLiang Fan, JinChang LiIn 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 semiparametric 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

Covariance kernel of linear spectral statistics for halfheavy tailed Wigner matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220912
Asad Lodhia, Anna MaltsevIn 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 αdependent expression for the covariance of the limiting process

Spectral measure of empirical autocovariance matrices of highdimensional Gaussian stationary processes Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220822
Arup Bose, Walid HachemConsider the empirical autocovariance matrices at given nonzero 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

Asymptotic freeness of unitary matrices in tensor product spaces for invariant states Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220810
Benoît Collins, Pierre Yves Gaudreau Lamarre, Camille MaleIn 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

Characteristic polynomials of random truncations: Moments, duality and asymptotics Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220728
Alexander Serebryakov, Nick Simm, Guillaume DubachWe 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

Limiting eigenvalue behavior of a class of large dimensional random matrices formed from a Hadamard product Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220728
Jack W. SilversteinThis 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

On the analytic structure of secondorder noncommutative probability spaces and functions of bounded Fréchet variation Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220722
Mario Diaz, James A. MingoIn 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 Poincarétype inequality for the linear statistics; and the existence of a secondorder limit distribution. This

Universal scaling limits of the symplectic elliptic Ginibre ensemble Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220722
SungSoo Byun, Markus EbkeWe 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 skeworthogonal 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 nonHermiticity

Threedimensional Gaussian fluctuations of spectra of overlapping stochastic Wishart matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220722
Jeffrey Kuan, Zhengye ZhouIn [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

Necessary and sufficient conditions for convergence to the semicircle distribution Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220714
Calvin Wooyoung ChinWe 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

On random matrices arising in deep neural networks: General I.I.D. case Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220714
Leonid Pastur, Victor SlavinWe 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

Operator level limit of the circular Jacobi βensemble Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220712
Yun Li, Benedek ValkóWe prove an operator level limit for the circular Jacobi βensemble. 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

Some characterization results on classical and free Poisson thinning Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220629
Soumendu Sundar MukherjeePoisson 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

The partition function of loggases with multiple odd charges Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220518
Elisha D. Wolff, Jonathan M. WellsWe use techniques in the shuffle algebra to present a formula for the partition function of a onedimensional loggas comprised of particles of (possibly) different integer charges at certain inverse temperature β in terms of the Berezin integral of an associated nonhomogeneous alternating tensor. This generalizes previously known results by removing the restriction on the number of species of odd

Edge fluctuations for random normal matrix ensembles Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220516
David GarcíaZeladaA famous result going back to Eric Kostlan states that the moduli of the eigenvalues of random normal matrices with radial potential are independent yet nonidentically 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

Large deviations for spectral measures of some spiked matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220504
Nathan Noiry, Alain RouaultWe 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.

Relating random matrix map enumeration to a universal symbol calculus for recurrence operators in terms of Bessel–Appell polynomials Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220429
Nicholas M. Ercolani, Patrick WatersMaps 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

Weak convergence of a collection of random functions defined by the eigenvectors of large dimensional random matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220423
Jack W. SilversteinFor 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](uki2−1n), 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

Adaptive singular value shrinkage estimate for low rank tensor denoising Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220419
Zerui Tao, Zhouping LiRecently, tensors are widely used to represent higherorder 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

Adaptive singular value shrinkage estimate for low rank tensor denoising Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220419
Zerui Tao, Zhouping LiRecently, tensors are widely used to represent higherorder 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

A new decomposition for multivalued 3 × 3 matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220412
Aymen Ammar, Aref Jeribi, Bilel SaadaouiIn 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.

High–low temperature dualities for the classical βensembles Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220405
Peter J. ForresterThe loop equations for the βensembles 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 onepoint 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

High–low temperature dualities for the classical βensembles Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220405
Peter J. ForresterThe loop equations for the βensembles 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 onepoint 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

On the operator norm of a Hermitian random matrix with correlated entries Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220319
Jana RekerWe 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.

Generalized heterogeneous hypergeometric functions and the distribution of the largest eigenvalue of an elliptical Wishart matrix Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220317
Aya Shinozaki, Koki Shimizu, Hiroki HashiguchiIn 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

Random matrices with independent entries: Beyond noncrossing partitions Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220308
Arup Bose, Koushik Saha, Arusharka Sen, Priyanka SenThe scaled standard Wigner matrix (symmetric with mean zero, variance one i.i.d. entries), and its limiting eigenvalue distribution, namely the semicircular distribution, have attracted much attention. The 2kth moment of the limit equals the number of noncrossing pairpartitions of the set {1, 2,…, 2k}. There are several extensions of this result in the literature. In this paper, we consider a unifying

Joint CLT for top eigenvalues of sample covariance matrices of separable high dimensional long memory processes Random Matrices Theory Appl. (IF 0.9) Pub Date : 20220308
Peng TianFor 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 semidefinite 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

Some strong convergence theorems for eigenvalues of general sample covariance matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 20211127
Yanqing YinThe 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

A dynamical version of the SYK model and the qBrownian motion Random Matrices Theory Appl. (IF 0.9) Pub Date : 20211122
Miguel Pluma, Roland SpeicherWe 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 qBrownian motion, a noncommutative 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

A dynamical version of the SYK model and the qBrownian motion Random Matrices Theory Appl. (IF 0.9) Pub Date : 20211122
Miguel Pluma, Roland SpeicherWe 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 qBrownian motion, a noncommutative 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

On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations Random Matrices Theory Appl. (IF 0.9) Pub Date : 20211117
Alicja DembczakKołodziejczyk, Anna LytovaGiven 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

On the asymptotic behavior of the eigenvalue distribution of block correlation matrices of highdimensional time series Random Matrices Theory Appl. (IF 0.9) Pub Date : 20211022
Philippe Loubaton, Xavier MestreWe consider linear spectral statistics built from the blocknormalized 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 crosscorrelation 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

Partial isometries, duality, and determinantal point processes Random Matrices Theory Appl. (IF 0.9) Pub Date : 20211022
Makoto Katori, Tomoyuki ShiraiA determinantal point process (DPP) is an ensemble of random nonnegativeintegervalued 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 L2spaces, L2(Sℓ,λℓ), ℓ=1,2, and introduce a bounded linear

Random Toeplitz matrices: The condition number under high stochastic dependence Random Matrices Theory Appl. (IF 0.9) Pub Date : 20211022
Paulo ManriqueMirónIn 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

On the asymptotic behavior of the eigenvalue distribution of block correlation matrices of highdimensional time series Random Matrices Theory Appl. (IF 0.9) Pub Date : 20211022
Philippe Loubaton, Xavier MestreWe consider linear spectral statistics built from the blocknormalized 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 crosscorrelation 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

Partial isometries, duality, and determinantal point processes Random Matrices Theory Appl. (IF 0.9) Pub Date : 20211022
Makoto Katori, Tomoyuki ShiraiA determinantal point process (DPP) is an ensemble of random nonnegativeintegervalued 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 L2spaces, L2(S ℓ,λℓ), ℓ = 1, 2, and introduce a bounded

Random Toeplitz matrices: The condition number under high stochastic dependence Random Matrices Theory Appl. (IF 0.9) Pub Date : 20211022
Paulo ManriqueMirónIn 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

Spectral properties for the Laplacian of a generalized Wigner matrix Random Matrices Theory Appl. (IF 0.9) Pub Date : 20211014
Anirban Chatterjee, Rajat Subhra HazraIn 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

The limit empirical spectral distribution of complex matrix polynomials Random Matrices Theory Appl. (IF 0.9) Pub Date : 20210925
Giovanni Barbarino, Vanni NoferiniWe study the empirical spectral distribution (ESD) for complex n × n matrix polynomials of degree k under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of

Gap probabilities in the bulk of the Airy process Random Matrices Theory Appl. (IF 0.9) Pub Date : 20210911
Elliot Blackstone, Christophe Charlier, Jonatan LenellsWe 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.

Global and local scaling limits for the β = 2 Stieltjes–Wigert random matrix ensemble Random Matrices Theory Appl. (IF 0.9) Pub Date : 20210811
Peter J. ForresterThe eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a wellknown analogy with the Boltzmann factor for a classical loggas with pair potential −log x − y, confined by a onebody harmonic potential. A generalization is to replace the pair potential by −log sinh(π(x − y)/L). The resulting PDF first appeared in the statistical physics literature in relation to nonintersecting

Regression conditions that characterize freePoisson and freeKummer distributions Random Matrices Theory Appl. (IF 0.9) Pub Date : 20210731
Agnieszka PiliszekWe 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 freeKummer and freePoisson (Marchenko–Pastur) noncommutative random variables.

Joint global fluctuations of complex Wigner and deterministic matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 20210716
Camile Male, James A. Mingo, Sandrine Péché, Roland SpeicherWe 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 secondorder and the limiting covariance depends the limiting ∗distribution of the deterministic matrices and their transposes and Hadamard products.

Asymptotic distribution of correlation matrix under blocked compound symmetric covariance structure Random Matrices Theory Appl. (IF 0.9) Pub Date : 20210701
Shinichi TsukadaAssuming 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

Moderate deviations for linear eigenvalue statistics of βensembles Random Matrices Theory Appl. (IF 0.9) Pub Date : 20210701
Fuqing Gao, Jianyong MuWe establish a moderate deviation principle for linear eigenvalue statistics of βensembles in the onecut regime with a realanalytic potential. The main ingredient is to obtain uniform estimates for the correlators of a family of perturbations of βensembles using the loop equations.

CLT with explicit variance for products of random singular matrices related to Hill’s equation Random Matrices Theory Appl. (IF 0.9) Pub Date : 20210701
Phanuel Mariano, Hugo PanzoWe 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 mdependent sequences

Applications in random matrix theory of a PIII′ τfunction sequence from Okamoto’s Hamiltonian formulation Random Matrices Theory Appl. (IF 0.9) Pub Date : 20210626
Dan Dai, Peter J. Forrester, ShuaiXia XuWe 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

Noncentral complex Wishart matrices: Moments and correlation of minors Random Matrices Theory Appl. (IF 0.9) Pub Date : 20210621
Velio Tralli, Andrea ContiComplex Wishart matrices are a class of random matrices with numerous emerging applications. In particular, the statistical characterization of such class of random matrices is essential for solving problems in various fields, including statistics, finance, physics and engineering. This paper establishes a new way to solve such problems based on the statistical moments and correlation of the minors

Largedimensional random matrix theory and its applications in deep learning and wireless communications Random Matrices Theory Appl. (IF 0.9) Pub Date : 20210618
Jungang Ge, YingChang Liang, Zhidong Bai, Guangming PanLargedimensional (LD) random matrix theory, RMT for short, which originates from the research field of quantum physics, has shown tremendous capability in providing deep insights into largedimensional systems. With the fact that we have entered an unprecedented era full of massive amounts of data and large complex systems, RMT is expected to play more important roles in the analysis and design of

Additivity violation of quantum channels via strong convergence to semicircular and circular elements Random Matrices Theory Appl. (IF 0.9) Pub Date : 20210618
Motohisa Fukuda, Takahiro Hasebe, Shinya SatoAdditivity violation of minimum output entropy, which shows nonclassical properties in quantum communication, had been proved in most cases for random quantum channels defined by Haardistributed unitary matrices. In this paper, we investigate random completely positive maps made of Gaussian Unitary Ensembles and Ginibre Ensembles regarding this matter. Using semicircular systems and circular systems