Symmetric Random Matrices Research Articles

Spectra of sparse random matrices

We compute the spectral density for ensembles of sparse symmetric random matrices using replica. Our formulation of the replica-symmetric ansatz shares the symmetries of that suggested in a seminal paper by Rodgers and Bray (symmetry with respect to permutation of replica and rotation symmetry in the space of replica), but uses a different representation in terms of superpositions of Gaussians. It gives rise to a pair of integral equations which can be solved by a stochastic population-dynamics algorithm. Remarkably our representation allows us to identify pure-point contributions to the spectral density related to the existence of normalizable eigenstates. Our approach is not restricted to matrices defined on graphs with Poissonian degree distribution. Matrices defined on regular random graphs or on scale-free graphs, are easily handled. We also look at matrices with row constraints such as discrete graph Laplacians. Our approach naturally allows us to unfold the total density of states into contributions coming from vertices of different local coordinations and an example of such an unfolding is presented. Our results are well corroborated by numerical diagonalization studies of large finite random matrices.

On the lower bound of the spectral norm of symmetric random matrices with independent entries

We show that the spectral radius of an $N\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from below by $ 2 \sigma - o( N^{-6/11+\varepsilon}), $ where $\sigma^2 $ is the variance of the matrix entries and $\varepsilon $ is an arbitrary small positive number. Combining with our previous result from [7], this proves that for any $\varepsilon <0, \ $ one has $ \|A_N\| =2 \sigma + o( N^{-6/11+\varepsilon}) $ with probability going to $ 1 $ as $N \to \infty$.

Spectral norm of random matrices

In this paper, we present a new upper bound for the spectral norm of symmetric random matrices with independent (but not necessarily identical) entries. Our results improve an earlier result of Furedi and Komlos.

Limit Theorems for Spectra of Random Matrices with Martingale Structure

We study classical ensembles of real symmetric random matrices introduced by Eugene Wigner. We discuss Stein’s method for the asymptotic approximation of expectations of functions of the normalized eigenvalue counting measure of high dimensional matrices. The method is based on a differential equation for the density of the semicircle law.

Eigenvalue density of cross-correlations in Sri Lankan financial market

We apply the universal properties with Gaussian orthogonal ensemble (GOE) of random matrices namely spectral properties, distribution of eigenvalues, eigenvalue spacing predicted by random matrix theory (RMT) to compare cross-correlation matrix estimators from emerging market data. The daily stock prices of the Sri Lankan All share price index and Milanka price index from August 2004 to March 2005 were analyzed. Most eigenvalues in the spectrum of the cross-correlation matrix of stock price changes agree with the universal predictions of RMT. We find that the cross-correlation matrix satisfies the universal properties of the GOE of real symmetric random matrices. The eigen distribution follows the RMT predictions in the bulk but there are some deviations at the large eigenvalues. The nearest-neighbor spacing and the next nearest-neighbor spacing of the eigenvalues were examined and found that they follow the universality of GOE. RMT with deterministic correlations found that each eigenvalue from deterministic correlations is observed at values, which are repelled from the bulk distribution.

Random symmetric matrices are almost surely nonsingular

Let Qn denote a random symmetric (n×n)-matrix, whose upper-diagonal entries are independent and identically distributed (i.i.d.) Bernoulli random variables (which take values 0 and 1 with probability 1/2). We prove that Qn is nonsingular with probability 1-O(n-1/8+δ) for any fixed δ>0. The proof uses a quadratic version of Littlewood-Offord-type results concerning the concentration functions of random variables and can be extended for more general models of random matrices

Sums of random symmetric matrices and quadratic optimization under orthogonality constraints

Let B i be deterministic real symmetric m × m matrices, and ξ i be independent random scalars with zero mean and “of order of one” (e.g., $$\xi_{i}\sim \mathcal{N}(0,1)$$). We are interested to know under what conditions “typical norm” of the random matrix $$S_N = \sum_{i=1}^N\xi_{i}B_{i}$$ is of order of 1. An evident necessary condition is $${\bf E}\{S_{N}^{2}\}\preceq O(1)I$$, which, essentially, translates to $$\sum_{i=1}^{N}B_{i}^{2}\preceq I$$; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of S N is $$\leq O(1)m^{{1\over 6}}$$: $${\rm Prob}\{||S_N||>\Omega m^{1/6}\}\leq O(1)\exp\{-O(1)\Omega^2\}$$ for all Ω > 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints $${\rm Opt} = \max\limits_{X_{j}\in{\bf R}^{m\times m}}\left\{F(X_1,\ldots ,X_k): X_jX_j^{\rm T}=I,\,j=1,\ldots ,k\right\}$$, where F is quadratic in X = (X 1,... ,X k ). We show that when F is convex in every one of X j , a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices $$X_j : {\rm Opt}\leq {\rm Opt}(SDP)\leq O(1) [m^{1/3}+\ln k]{\rm Opt}$$.

Multivariate Matsumoto–Yor property is rather restrictive

Matsumoto and Yor [2001. An analogue of Pitman's 2 M - X theorem for exponential Wiener functionals. Part II: the role of the GIG laws. Nagoya Math. J. 162, 65–86] discovered an interesting invariance property of a product of the generalized inverse Gaussian (GIG) and the gamma distributions. For univariate random variables or symmetric positive definite random matrices it is a characteristic property for this pair of distributions. It appears that for random vectors the Matsumoto–Yor property characterizes only very special families of multivariate GIG and gamma distributions: components of the respective random vectors are grouped into independent subvectors, each subvector having linearly dependent components. This complements the version of the multivariate Matsumoto–Yor property on trees and related characterization obtained in Massam and Wesołowski [2004. The Matsumoto–Yor property on trees. Bernoulli 10, 685–700].

Spectral measure of large random Hankel, Markov and Toeplitz matrices

We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables {Xk} of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {Xij}j>i of zero mean and unit variance, scaling the eigenvalues by $\sqrt{n}$ we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions γH, γM and γT of unbounded support. The moments of γH and γT are the sum of volumes of solids related to Eulerian numbers, whereas γM has a bounded smooth density given by the free convolution of the semicircle and normal densities. For symmetric Markov matrices generated by i.i.d. random variables {Xij}j>i of mean m and finite variance, scaling the eigenvalues by n we prove the almost sure, weak convergence of the spectral measures to the atomic measure at −m. If m=0, and the fourth moment is finite, we prove that the spectral norm of Mn scaled by $\sqrt{2n\log n}$ converges almost surely to 1.

Universality of the Edge Distribution of Eigenvalues of Wigner Random Matrices with Polynomially Decaying Distributions of Entries

We consider an ensemble of Wigner symmetric random matrices An={aij}, i,j=1, . . . ,n with matrix elements aij, being i.i.d. symmetrically distributed random variables Open image in new window We assume that Open image in new window and that

Real symmetric random matrices and path counting

Exact evaluation of (TrS(p)) is here performed for real symmetric matrices S of arbitrary order n , up to some integer p , where the matrix entries are independent identically distributed random variables, with an arbitrary probability distribution. These expectations are polynomials in the moments of the matrix entries; they provide useful information on the spectral density of the ensemble in the large n limit. They also are a straightforward tool to examine a variety of rescalings of the entries in the large n limit.

A CLT for a band matrix model

A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poincare type. Special cases treated, with an explicit evaluation of limiting variances, are generalized Wigner and Wishart matrices.

Statistics of the eigenvalues of Tsallis matrices

The statistics of the eigenvalues of symmetric random matrices, composed by real and statistically independent elements following the distribution that maximizes Tsallis's entropy, is carried numerically in the limit of large matrices. For entropic indexes in the interval −∞<q< 5 3 , by using a convenient rescale of variables, it is possible to show that such matrices fall in the same class of the Gaussian orthogonal ensemble (GOE). For the entropic index 5 3 ⩽q<3 , the density of eigenvalues and the distribution of level spacings do not seem to follow a simple rescale of variables involving different values of q, and exhibit a behavior very distinct from the GOE: both quantities present long tails for 5 3 ⩽q⩽2 , and such long tails die out when q>2. The density of eigenvalues appears to be symmetric around zero, exhibiting a peak at the origin that becomes steeper for increasing values of q, approaching a delta function at the origin when q→3. The distribution of level spacings displays a form that resembles the well-known Wigner's surmise (augmented by a long tail) for q slightly greater than 5 3 , but gets deformed for increasing values of q, approaching an exponential decay for q>2. For q close to 3, our results resemble those of very sparse random matrices (characterized by many zero matrix elements).

Random symmetric matrices with a constraint: the spectral density of random impedance networks.

We derive the mean eigenvalue density for symmetric Gaussian random N x N matrices in the limit of large N, with a constraint implying that the row sum of matrix elements should vanish. The result is shown to be equivalent to a result found recently for the average density of resonances in random impedance networks [Y.V. Fyodorov, J. Phys. A 32, 7429 (1999)]. In the case of banded matrices, the analytical results are compared with those extracted from the numerical solution of Kirchhoff equations for quasi-one-dimensional random impedance networks.

Random matrix theory and non-parametric model of random uncertainties in vibration analysis

Recently, a new approach, called a non-parametric model of random uncertainties, has been introduced for modelling random uncertainties in linear and non-linear elastodynamics in the low-frequency range. This non-parametric approach differs from the parametric methods for random uncertainties modelling and has been developed in introducing a new ensemble of random matrices constituted of symmetric positive-definite real random matrices. This ensemble differs from the Gaussian orthogonal ensemble (GOE) and from the other known ensembles of the random matrix theory. The present paper has three main objectives. The first one is to study the statistics of the random eigenvalues of random matrices belonging to this new ensemble and to compare with the GOE. The second one is to compare this new ensemble of random matrices with the GOE in the context of the non-parametric approach of random uncertainties in structural dynamics for the low-frequency range. The last objective is to give a new validation for the non-parametric model of random uncertainties in structural dynamics in comparing, in the low-frequency range, the dynamical response of a simple system having random uncertainties modelled by the parametric and the non-parametric methods. These three objectives will allow us to conclude about the validity of the different theories.

Random Uncertainties Model in Dynamic Substructuring Using a Nonparametric Probabilistic Model

This paper presents a new approach, called a nonparametric approach, for constructing a model of random uncertainties in dynamic substructuring in order to predict the matrix-valued frequency response functions of complex structures. Such an approach allows nonhomogeneous uncertainties to be modeled with the nonparametric approach. The Craig-Bampton dynamic substructuring method is used. For each substructure, a nonparametric model of random uncertainties is introduced. This nonparametric model does not require identifying uncertain parameters in the reduced matrix model of each substructure as is usually done for the parametric approach. This nonparametric model of random uncertainties is based on the use of a probability model for symmetric positive-definite real random matrices using the entropy optimization principle. The theory and a numerical example are presented in the context of the finite-element method. The numerical results obtained show the efficiency of the model proposed.

Products of correlated symmetric matrices and q-Catalan numbers

The well known convergence of the spectrum of large random symmetric matrices, due to Wigner, holds for products of correlated symmetric matrices with general entries. The limiting moments coincide with weighted enumeration of permutations, or of rooted trees. When the correlations are Markovian, the limiting first moments are closely related to Carlitz-Riordan q-Catalan numbers. As a consequence, these moments asymptotically exhibit a phase transition, with respect to the correlation coefficient. The critical correlations can be computed as the least positive zero of q-hypergeometric functions. Similar methods allow to recover some results due to Logan, Mazo, Odlyzko and Shepp.

Generalized Wigner surmise for (2×2) random matrices

We present new analytical results concerning the spectral distributions for (2×2) random real symmetric matrices which generalize the Wigner surmise.

Two Krylov space algorithms for repeated large scale sparse matrix diagonalization

Two simple algorithms for the diagonalization of a set of sparse symmetric matrices of the form A+{Δk}k=1L for large values of L are proposed and investigated. The numerical strategies economize computer resources by requiring the reconstruction of the Lanczos basis for a small number of times compared to L. Each member of the set {Δk}k=1L is assumed to have a smaller number of nonzero elements compared to A. Both numerical procedures are derived from the Lanczos algorithm and use periodically a recursion to obtain the Lanczos vectors. Tests are conducted with both random symmetric matrices and with DVR Hamiltonians containing parametric potentials. The performance of the algorithms in terms of numerical accuracy, stability, and CPU time is studied as a function of two properties of the matrix set {Δk}k=1L.

Extended proof of the Statement: Convergence rate of the expected spectral functions of symmetric random matrices Ξ n is equal to O (n—1/2) and the method of critical steepest descent

Extended proof of the Statement: Convergence rate of the expected spectral functions of symmetric random matrices Ξ n is equal to O (n—1/2) and the method of critical steepest descent