-
A degenerate Gaussian weight connected with Painlevé equations and Heun equations Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-12-26 Pengju Han; Yang Chen
In this paper, we study the recurrence coefficients of a deformed Hermite polynomials orthogonal with respect to the weight w(x;t,α):=e−x2|x−t|α(A+B⋅𝜃(x−t)),x∈(−∞,∞), where α>−1,A≥0,A+B≥0 and t∈ℝ. It is an extension of Chen and Feigin [J. Phys. A., Math. Gen. 39 (2006) 12381–12393]. By using the ladder operator technique, we show that the recurrence coefficients satisfy a particular Painlevé IV equation
-
Pair dependent linear statistics for CβE Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-12-17 Ander Aguirre; Alexander Soshnikov; Joshua Sumpter
We study the limiting distribution of a pair counting statistics of the form ∑1≤i≠j≤Nf(LN(𝜃i−𝜃j)) for the circular β-ensemble (CβE) of random matrices for sufficiently smooth test function f and LN=O(N). For β=2 and LN=N our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.
-
Boolean cumulants and subordination in free probability Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-12-17 Franz Lehner; Kamil Szpojankowski
Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation 𝔼φ(z−X−f(X)Yf∗(X))−1|X for free random variables X,Y and a Borel function f is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form X+f(X)Yf∗(X)
-
Some patterned matrices with independent entries Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-11-27 Arup Bose; Koushik Saha; Priyanka Sen
Patterned random matrices such as the reverse circulant, the symmetric circulant, the Toeplitz and the Hankel matrices and their almost sure limiting spectral distribution (LSD), have attracted much attention. Under the assumption that the entries are taken from an i.i.d. sequence with finite variance, the LSD are tied together by a common thread — the 2kth moment of the limit equals a weighted sum
-
Uniform point variance bounds in classical beta ensembles Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-11-27 Joseph Najnudel; Bálint Virág
In this paper, we give bounds on the variance of the number of points of the Circular and the Gaussian β Ensemble in arcs of the unit circle or intervals of the real line. These bounds are logarithmic with respect to the renormalized length of these sets, which is expected to be optimal up to a multiplicative constant depending only on β.
-
Spectrum of SYK model II: Central limit theorem Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-11-27 Renjie Feng; Gang. Tian; Dongyi. Wei
In our previous paper [R. Feng, G. Tian and D. Wei, Spectrum of SYK model, Peking Math. J.2 (2019) 41–70], we derived the almost sure convergence of the global density of eigenvalues of random matrices of the SYK model. In this paper, we will prove the central limit theorem for the linear statistics of eigenvalues of the SYK model and compute its variance.
-
Finite free convolutions via Weingarten calculus Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-11-27 Jacob Campbell; Zhi Yin
We consider the three finite free convolutions for polynomials studied in a recent paper by Marcus, Spielman and Srivastava. Each can be described either by direct explicit formulae or in terms of operations on randomly rotated matrices. We present an alternate approach to the equivalence between these descriptions, based on combinatorial Weingarten methods for integration over the unitary and orthogonal
-
Polynomial with cyclic monotone elements with applications to Random Matrices with discrete spectrum Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-11-05 Octavio Arizmendi; Adrián Celestino
We provide a generalization and new proofs of the formulas of Collins et al. for the spectrum of polynomials in cyclic monotone elements. This is applied to Random Matrices with discrete spectrum.
-
Process convergence of fluctuations of linear eigenvalue statistics of random circulant matrices Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-11-05 Arup Bose; Shambhu Nath Maurya; Koushik Saha
We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of random circulant matrices with independent Brownian motion entries, as the dimension of the matrix tends to ∞. Our derivation is based on the trace formula of circulant matrix, method of moments and some combinatorial techniques.
-
The strong circular law: A combinatorial view Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-11-05 Vishesh Jain
Let Nn be an n×n complex random matrix, each of whose entries is an independent copy of a centered complex random variable z with finite nonzero variance σ2. The strong circular law, proved by Tao and Vu, states that almost surely, as n→∞, the empirical spectral distribution of Nn/(σn) converges to the uniform distribution on the unit disc in ℂ. A crucial ingredient in the proof of Tao and Vu, which
-
Rate of convergence to the Circular Law via smoothing inequalities for log-potentials Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-09-24 Friedrich Götze; Jonas Jalowy
The aim of this paper is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by n−1/2. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration
-
Differential and difference equations for recurrence coefficients of orthogonal polynomials with hypergeometric weights and Bäcklund transformations of the sixth Painlevé equation Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-09-24 Jie hu; Galina Filipuk; Yang Chen
It is known from [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl.14 (2018), Article ID: 088, 19 pp.] that the recurrence coefficients of discrete orthogonal polynomials on the nonnegative integers with hypergeometric weights satisfy a system of nonlinear difference equations. There is also a connection to
-
Equivalent symmetric kernels of determinantal point processes Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-09-19 Marco Stevens
Determinantal point processes are point processes whose correlation functions are given by determinants of matrices. The entries of these matrices are given by one fixed function of two variables, which is called the kernel of the point process. It is well known that there are different kernels that induce the same correlation functions. We classify all the possible transformations of a kernel that
-
Eigenvalues of the non-backtracking operator detached from the bulk Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-09-19 Simon Coste; Yizhe Zhu
We describe the non-backtracking spectrum of a stochastic block model with connection probabilities pin,pout=ω(logn)/n. In this regime we answer a question posed in [L. Dall’Amico, R. Couillet and N. Tremblay, Revisiting the Bethe–Hessian: Improved community detection in sparse heterogeneous graphs, in Advances in Neural Information Processing Systems (2019), pp. 4039–4049] regarding the existence
-
Fluctuations of the spectrum in rotationally invariant random matrix ensembles Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-09-10 Elizabeth S. Meckes; Mark W. Meckes
We investigate traces of powers of random matrices whose distributions are invariant under rotations (with respect to the Hilbert–Schmidt inner product) within a real-linear subspace of the space of n×n matrices. The matrices, we consider may be real or complex, and Hermitian, antihermitian, or general. We use Stein’s method to prove multivariate central limit theorems, with convergence rates, for
-
Random non-Abelian G-circulant matrices. Spectrum of random convolution operators on large finite groups Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-08-19 Radosław Adamczak
We analyze the limiting behavior of the eigenvalue and singular value distribution for random convolution operators on large (not necessarily Abelian) groups, extending the results by Meckes for the Abelian case. We show that for regular sequences of groups, the limiting distribution of eigenvalues (respectively singular values) is a mixture of eigenvalue (respectively singular value) distributions
-
A semiparametric model for matrix regression Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-08-12 Weihua Zhao; Xiaoyu Zhang; Heng Lian
We focus on regression problems in which the predictors are naturally in the form of matrices. Reduced rank regression and related regularized method have been adapted to matrix regression. However, linear methods are restrictive in their expressive power. In this work, we consider a class of semiparametric additive models based on series estimation of nonlinear functions which interestingly induces
-
On the behavior of large empirical autocovariance matrices between the past and the future Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-07-28 P. Loubaton; D. Tieplova
The asymptotic behavior of the distribution of the squared singular values of the sample autocovariance matrix between the past and the future of a high-dimensional complex Gaussian uncorrelated sequence is studied. Using Gaussian tools, it is established that the distribution behaves as a deterministic probability measure whose support 𝒮 is characterized. It is also established that the squared singular
-
Linear differential equations for the resolvents of the classical matrix ensembles Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-07-15 Anas A. Rahman; Peter J. Forrester
The spectral density for random matrix β ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of β, which for even β is a polynomial of degree β(N−1). In the cases of the classical Gaussian, Laguerre, and Jacobi weights, we show that this polynomial, and moreover, the spectral density itself, can be characterized as the solution
-
Linear eigenvalue statistics of random matrices with a variance profile Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-07-06 Kartick Adhikari; Indrajit Jana; Koushik Saha
We give an upper bound on the total variation distance between the linear eigenvalue statistic, properly scaled and centered, of a random matrix with a variance profile and the standard Gaussian random variable. The second-order Poincaré inequality-type result introduced in [S. Chatterjee, Fluctuations of eigenvalues and second order poincaré inequalities, Prob. Theory Rel. Fields143(1) (2009) 1–40
-
Efficient estimation of reduced-rank partial envelope model in multivariate linear regression Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-06-24 Jing Zhang; Zhensheng Huang; Yan Xiong
In order to further improve the efficiency of parameter estimation and reduce the number of estimated parameters, we adopt dimension reduction ideas of partial envelope model proposed by [Su and Cook, Partial envelopes for efficient estimation in multivariate linear regression, Biometrika98 (2011) 133–146.] to center on some predictors of special interest. Based on the research results of [Cook et
-
Tracy–Widom law for the largest eigenvalue of sample covariance matrix generated by VARMA Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-05-28 Boping Tian; Yangchun Zhang; Wang Zhou
In this paper, we derive the Tracy–Widom law for the largest eigenvalue of sample covariance matrix generated by the vector autoregressive moving average model when the dimension is comparable to the sample size. This result is applied to make inference on the vector autoregressive moving average model. Simulations are conducted to demonstrate the finite sample performance of our inference.
-
Eigenvectors of a matrix under random perturbation Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-05-28 Florent Benaych-Georges; Nathanaël Enriquez; Alkéos Michaïl
In this text, based on elementary computations, we provide a perturbative expansion of the coordinates of the eigenvectors of a Hermitian matrix of large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent, centered, with a variance profile. This is done through a perturbative expansion of spectral measures associated to
-
Quaternionic second-order freeness and the fluctuations of large symplectically invariant random matrices Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-04-28 C. E. I. Redelmeier
We present a definition for second-order freeness in the quaternionic case. We demonstrate that this definition on a second-order probability space is asymptotically satisfied by independent symplectically invariant quaternionic matrices. This definition is different from the natural definition for complex and real second-order probability spaces, those motivated by the asymptotic behavior of unitarily
-
Moments of the ground state density for the d-dimensional Fermi gas in an harmonic trap Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-04-21 Peter J. Forrester
We consider properties of the ground state density for the d-dimensional Fermi gas in an harmonic trap. Previous work has shown that the d-dimensional Fourier transform has a very simple functional form. It is shown that this fact can be used to deduce that the density itself satisfies a third-order linear differential equation, previously known in the literature but from other considerations. It is
-
Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-03-20 Theodoros Assiotis; Jonathan P. Keating
In this note, we give a combinatorial and noncomputational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity. This is achieved by relating these quantities to a lattice point count problem. Our main result is a new explicit expression for the leading order coefficient in the
-
Harmonic means of Wishart random matrices Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-02-19 Asad Lodhia
We use free probability to compute the limiting spectral properties of the harmonic mean of n i.i.d. Wishart random matrices Wi whose limiting aspect ratio is γ∈(0,1) when 𝔼[Wi]=I. We demonstrate an interesting phenomenon where the harmonic mean H of the n Wishart matrices is closer in operator norm to 𝔼[Wi] than the arithmetic mean A for small n, after which the arithmetic mean is closer. We also
-
Universality for conditional measures of the Bessel point process Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-02-06 Leslie D. Molag; Marco Stevens
The Bessel point process is a rigid point process on the positive real line and its conditional measure on a bounded interval [0,R] is almost surely an orthogonal polynomial ensemble. In this paper, we show that if R tends to infinity, one almost surely recovers the Bessel point process. In fact, we show this convergence for a deterministic class of probability measures, to which the conditional measure
-
Spiked sample covariance matrices with possibly multiple bulk components Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-02-06 Xiucai Ding
In this paper, we study the convergent limits and rates of the eigenvalues and eigenvectors for spiked sample covariance matrices whose spectrum can have multiple bulk components. Our model is an extension of Johnstone’s spiked covariance matrix model. Based on our results, we can extend many statistical applications based on Johnstone’s spiked covariance matrix model.
-
Eigenvalue rigidity for truncations of random unitary matrices Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-02-06 Elizabeth Meckes; Kathryn Stewart
We consider the empirical eigenvalue distribution of an m×m principal submatrix of an n×n random unitary matrix distributed according to Haar measure. For n and m large with mn=α, the empirical spectral measure is well approximated by a deterministic measure μα supported on the unit disc. In earlier work, we showed that for fixed n and m, the bounded-Lipschitz distance dBL between the empirical spectral
-
Symmetries of the quaternionic Ginibre ensemble Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-01-10 Guillaume Dubach
We establish a few properties of eigenvalues and eigenvectors of the quaternionic Ginibre ensemble (QGE), analogous to what is known in the complex Ginibre case (see [7, 11, 14]). We first recover a version of Kostlan’s theorem that was already at the heart of an argument by Rider [1], namely, that the set of the squared radii of the eigenvalues is distributed as a set of independent gamma variables
-
Large deviations and a new sum rule for spectral matrix measures of the Jacobi ensemble Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-12-20 Fabrice Gamboa; Jan Nagel; Alain Rouault
We continue to explore the connections between large deviations for objects coming from random matrix theory and sum rules. This connection was established in [Sum rules via large deviations, J. Funct. Anal. 270(2) (2016) 509–559] for spectral measures of classical ensembles (Gauss–Hermite, Laguerre, Jacobi) and it was extended to spectral matrix measures of the Hermite and Laguerre ensemble in [Sum
-
Spectral distributions of periodic random matrix ensembles Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-12-19 Roger Van Peski
Koloğlu, Kopp and Miller compute the limiting spectral distribution of a certain class of real random matrix ensembles, known as k-block circulant ensembles, and discover that it is exactly equal to the eigenvalue distribution of an k×k Gaussian unitary ensemble. We give a simpler proof that under very general conditions which subsume the cases studied by Koloğlu–Kopp–Miller, real-symmetric ensembles
-
On the distribution of an arbitrary subset of the eigenvalues for some finite dimensional random matrices Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-12-12 Marco Chiani; Alberto Zanella
We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Specifically, we derive compact expressions for the joint probability distribution function of the eigenvalues and the expectation of functions of the eigenvalues
-
Fine asymptotics for models with Gamma type moments Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-12-09 Peter Eichelsbacher; Lukas Knichel
The aim of this paper is to give fine asymptotics for random variables with moments of Gamma type. Among the examples, we consider random determinants of Laguerre and Jacobi beta ensembles with varying dimensions (the number of observed variables and the number of measurements vary and may be different). In addition to the Dyson threefold way of classical random matrix models (GOE, GUE, GSE), we study
-
Spectra of adjacency and Laplacian matrices of inhomogeneous Erdős–Rényi random graphs Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-12-04 Arijit Chakrabarty; Rajat Subhra Hazra; Frank den Hollander; Matteo Sfragara
This paper considers inhomogeneous Erdős–Rényi random graphs 𝔾N on N vertices in the non-sparse non-dense regime. The edge between the pair of vertices {i,j} is retained with probability 𝜀Nf(iN,jN), 1≤i≠j≤N, independently of other edges, where f:[0,1]×[0,1]→[0,∞) is a continuous function such that f(x,y)=f(y,x) for all x,y∈[0,1]. We study the empirical distribution of both the adjacency matrix AN
-
The densities and distributions of the largest eigenvalue and the trace of a Beta–Wishart matrix Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-12-03 Vesselin Drensky; Alan Edelman; Tierney Genoar; Raymond Kan; Plamen Koev
We present new expressions for the densities and distributions of the largest eigenvalue and the trace of a Beta–Wishart matrix. The series expansions for these expressions involve fewer terms than previously known results. For the trace, we also present a new algorithm that is linear in the size of the matrix and the degree of truncation, which is optimal.
-
Approximation of the power functions of Roy’s largest root test under general spiked alternatives Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-11-26 Zhiqiang Hou; Yan Liu; Zhidong Bai; Jiang Hu
Roy’s largest root is a common test statistic in multivariate analysis, statistical signal processing and related fields. According to Anderson [An Introduction to Multivariate Statistical Analysis, 3rd edn. (Wiley, New York, 2003)], it is numerically known that compared with the other three tests of linear hypotheses, Roy’s largest root test has the highest power under rank-one alternatives. Therefore
-
Local laws for non-Hermitian random matrices and their products Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-11-15 Friedrich Götze; Alexey Naumov; Alexander Tikhomirov
We consider products of independent n×n non-Hermitian random matrices X(1),…,X(m). Assume that their entries, Xjk(q),1≤j,k≤n,q=1,…,m, are independent identically distributed random variables with zero mean, unit variance. Götze and Tikhomirov [On the asymptotic spectrum of products of independent random matrices, preprint (2010), arXiv:1012.2710] and O’Rourke and Sochnikov [Products of independent
-
Additive matrix convolutions of Pólya ensembles and polynomial ensembles Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-11-08 Mario Kieburg
Recently, subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called Pólya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the respective convolutions and, thus, build a semi-group when adding by hand a unit element. They even have a semi-group action on the polynomial ensembles. Moreover, in several
-
Estimation and testing for panel data partially linear single-index models with errors correlated in space and time Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-11-07 Jian-Qiang Zhao; Yan-Yong Zhao; Jin-Guan Lin; Zhang-Xiao Miao; Waled Khaled
We consider a panel data partially linear single-index models (PDPLSIM) with errors correlated in space and time. A serially correlated error structure is adopted for the correlation in time. We propose using a semiparametric minimum average variance estimation (SMAVE) to obtain estimators for both the parameters and unknown link function. We not only establish an asymptotically normal distribution
-
Robust feature screening for multi-response trans-elliptical regression model with ultrahigh-dimensional covariates Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-11-04 Yong He; Hao Sun; Jiadong Ji; Xinsheng Zhang
In this paper, we innovatively propose an extremely flexible semi-parametric regression model called Multi-response Trans-Elliptical Regression (MTER) Model, which can capture the heavy-tail characteristic and tail dependence of both responses and covariates. We investigate the feature screening procedure for the MTER model, in which Kendall’ tau-based canonical correlation estimators are proposed
-
No eigenvalues outside the support of the limiting spectral distribution of quaternion sample covariance matrices Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-11-04 Huiqin Li
In this paper, we consider the spectral properties of quaternion sample covariance matrices. Let Bn=1nTn1/2XnXn∗Tn1/2, where Tn1/2 is the square root of a p×p quaternion Hermitian non-negative definite matrix Tn and Xn is a p×n matrix consisting of i.i.d. standard quaternion entries. Under the framework of random matrix theory, i.e. p/n→c∈(0,∞) as n→∞, we prove that if the fourth moment of the entries
-
Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-10-23 Peter J. Forrester; Jesper R. Ipsen; Dang-Zheng Liu; Lun Zhang
In this paper, we highlight the role played by orthogonal and symplectic Harish-Chandra integrals in the study of real-valued matrix product ensembles. By making use of these integrals and the matrix-valued Fourier-Laplace transform, we find the explicit eigenvalue distributions for particular Hermitian anti-symmetric matrices and Hermitian anti-self dual matrices, involving both sums and products
-
Multiple change-points detection in high dimension Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-10-23 Yunlong Wang; Changliang Zou; Zhaojun Wang; Guosheng Yin
Change-point detection is an integral component of statistical modeling and estimation. For high-dimensional data, classical methods based on the Mahalanobis distance are typically inapplicable. We propose a novel testing statistic by combining a modified Euclidean distance and an extreme statistic, and its null distribution is asymptotically normal. The new method naturally strikes a balance between
-
Painlevé V, Painlevé XXXIV and the degenerate Laguerre unitary ensemble Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-10-14 Chao Min; Yang Chen
In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble (dLUE). This problem originates from the largest or smallest eigenvalue distribution of the dLUE. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight function. By applying the ladder operators to our problem, we obtain two auxiliary quantities
-
Empirical likelihood for high-dimensional partially functional linear model Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-09-11 Zhiqiang Jiang; Zhensheng Huang; Guoliang Fan
This paper considers empirical likelihood inference for a high-dimensional partially functional linear model. An empirical log-likelihood ratio statistic is constructed for the regression coefficients of non-functional predictors and proved to be asymptotically normally distributed under some regularity conditions. Moreover, maximum empirical likelihood estimators of the regression coefficients of
-
Collision of eigenvalues for matrix-valued processes Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-08-23 Arturo Jaramillo; David Nualart
We examine the probability that at least two eigenvalues of a Hermitian matrix-valued Gaussian process, collide. In particular, we determine sharp conditions under which such probability is zero. As an application, we show that the eigenvalues of a real symmetric matrix-valued fractional Brownian motion of Hurst parameter H, collide when H<1/2 and do not collide when H>12, while those of a complex
-
On the determinantal structure of conditional overlaps for the complex Ginibre ensemble Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-08-01 Gernot Akemann; Roger Tribe; Athanasios Tsareas; Oleg Zaboronski
We continue the study of joint statistics of eigenvectors and eigenvalues initiated in the seminal papers of Chalker and Mehlig. The principal object of our investigation is the expectation of the matrix of overlaps between the left and the right eigenvectors for the complex N×N Ginibre ensemble, conditional on an arbitrary number k=1,2,… of complex eigenvalues. These objects provide the simplest generalization
-
Boundaries of sine kernel universality for Gaussian perturbations of Hermitian matrices Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-07-26 Tom Claeys; Thorsten Neuschel; Martin Venker
We explore the boundaries of sine kernel universality for the eigenvalues of Gaussian perturbations of large deterministic Hermitian matrices. Equivalently, we study for deterministic initial data the time after which Dyson’s Brownian motion exhibits sine kernel correlations. We explicitly describe this time span in terms of the limiting density and rigidity of the initial points. Our main focus lies
-
The generating function for the Bessel point process and a system of coupled Painlevé V equations Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-07-26 Christophe Charlier; Antoine Doeraene
We study the joint probability generating function for k occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painlevé V equations, which are derived from a Lax pair of a Riemann–Hilbert problem. This generalizes a result of Tracy and Widom [C. A. Tracy
-
Bounds on the norm of Wigner-type random matrices Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-07-26 László Erdős; Peter Mühlbacher
We consider a Wigner-type ensemble, i.e. large hermitian N×N random matrices H=H∗ with centered independent entries and with a general matrix of variances Sxy=𝔼|Hxy|2. The norm of H is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of S that substantially improves the earlier bound 2∥S∥∞1/2
-
Adjacency matrix comparison for stochastic block models Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-07-26 Guangren Yang; Songshan Yang; Wang Zhou
In this paper, we study whether two networks arising from two stochastic block models have the same connection structures by comparing their adjacency matrices. We conduct Monte Carlo simulations study to examine the finite sample performance of the proposed method. A real data example is used to illustrate the proposed methodology.
-
Singular values of large non-central random matrices Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-07-26 Włodek Bryc; Jack W. Silverstein
We study largest singular values of large random matrices, each with mean of a fixed rank K. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It provides a decomposition of the largest K singular values into the deterministic rate of growth, random centered fluctuations given as explicit linear combinations of
-
Nonuniversality of fluctuations of outliers for Hermitian polynomials in a complex Wigner matrix and a spiked diagonal matrix Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-07-12 Mireille Capitaine
We study the fluctuations associated to the a.s. convergence of the outliers established by Belinschi–Bercovici–Capitaine of an Hermitian polynomial in a complex Wigner matrix and a spiked deterministic real diagonal matrix. Thus, we extend the nonuniversality phenomenon established by Capitaine–Donati-Martin–Féral for additive deformations of complex Wigner matrices, to any Hermitian polynomial. The
-
Spectral densities of singular values of products of Gaussian and truncated unitary random matrices Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-07-12 Thorsten Neuschel
We study the densities of limiting distributions of squared singular values of high-dimensional matrix products composed of independent complex Gaussian (complex Ginibre) and truncated unitary matrices which are taken from Haar distributed unitary matrices with appropriate dimensional growth. In the general case, we develop a new approach to obtain complex integral representations for densities of
-
The joint distribution of the marginals of multipartite random quantum states Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-07-04 Stephane Dartois; Luca Lionni; Ion Nechita
We study the joint distribution of the set of all marginals of a random Wishart matrix acting on a tensor product Hilbert space. We compute the limiting free mixed cumulants of the marginals, and we show that in the balanced asymptotical regime, the marginals are asymptotically free. We connect the matrix integrals relevant to the study of operators on tensor product spaces with the corresponding classes
-
Outlier detection for multinomial data with a large number of categories Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-06-18 Xiaona Yang; Zhaojun Wang; Xuemin Zi
This paper develops an outlier detection procedure for multinomial data when the number of categories tends to infinity. Most of the outlier detection methods are based on the assumption that the observations follow multivariate normal distribution, while in many modern applications, the observations either are measured on a discrete scale or naturally have some categorical structures. For such multinomial
-
A differential equation for varying Krall-type orthogonal polynomials Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-06-11 Galina Filipuk; Juan F. Mañas-Mañas
In this contribution, we consider varying Krall-type polynomials which are orthogonal with respect to a varying discrete Krall-type inner product. Our main goal is to give ladder operators for this family of polynomials as well as to find a second-order differential-difference equation that these polynomials satisfy. We generalize some results that appeared recently in the literature.
-
Matrix factorizations and orthogonal polynomials Random Matrices Theory Appl. (IF 1.206) Pub Date : 2019-06-11 Diego Dominici
We present some elements of the theory of orthogonal polynomials based on matrix decompositions. We focus our attention on discrete linear functionals, and use the Meixner polynomials as a concrete example.
Contents have been reproduced by permission of the publishers.