Statistics Of Random Matrices Research Articles

Central limit theorem for linear eigenvalue statistics of random matrices with independent entries

We consider $n\times n$ real symmetric and Hermitian Wigner random matrices $n^{-1/2}W$ with independent (modulo symmetry condition) entries and the (null) sample covariance matrices $n^{-1}X^*X$ with independent entries of $m\times n$ matrix $X$. Assuming first that the 4th cumulant (excess) $\kappa_4$ of entries of $W$ and $X$ is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as $n\to\infty$, $m\to\infty$, $m/n\to c\in[0,\infty)$ with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class $\mathbf{C}^5$). This is done by using a simple ``interpolation trick'' from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially $\mathbb{C}^5$ test function. Here the variance of statistics contains an additional term proportional to $\kappa_4$. The proofs of all limit theorems follow essentially the same scheme.

Fluctuations of Matrix Elements of Regular Functions of Gaussian Random Matrices

We find the limit of the variance and prove the Central Limit Theorem (CLT) for the matrix elements φjk(M), j,k=1,…,n of a regular function φ of the Gaussian matrix M (GOE and GUE) as its size n tends to infinity. We show that unlike the linear eigenvalue statistics Tr φ(M), a traditional object of random matrix theory, whose variance is bounded as n→∞ and the CLT is valid for Tr φ(M)−E{Tr φ(M)}, the variance of φjk(M) is O(1/n), and the CLT is valid for \(\sqrt{n}(\varphi _{jk}(M)-\mathbf{E}\{\varphi _{jk}(M)\})\) . This shows the role of eigenvectors in the forming of the asymptotic regime of various functions (statistics) of random matrices. Our proof is based on the use of the Fourier transform as a basic characteristic function, unlike the Stieltjes transform and moments, used in majority of works of the field. We also comment on the validity of analogous results for other random matrices.

A CLT for information-theoretic statistics of Gram random matrices with a given variance profile

Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $$ Y_{ij}^{n}=\frac{\sigma_{ij}(n)}{\sqrt{n}} X_{ij}^{n} , $$ the $X_{ij}^{n}$ being centered, independent and identically distributed random variables with unit variance and $(\sigma_{ij}(n); 1\le i\le N, 1\le j\le n)$ being an array of numbers we shall refer to as a variance profile. We study in this article the fluctuations of the random variable $$ \log\det\left(Y_n Y_n^* + \rho I_N \right) $$ where $Y^*$ is the Hermitian adjoint of $Y$ and $\rho > 0$ is an additional parameter. We prove that when centered and properly rescaled, this random variable satisfies a Central Limit Theorem (CLT) and has a Gaussian limit whose parameters are identified. A complete description of the scaling parameter is given; in particular it is shown that an additional term appears in this parameter in the case where the 4$^\textrm{th}$ moment of the $X_{ij}$'s differs from the 4$^{\textrm{th}}$ moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications.

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.

Virial expansion for almost diagonal random matrices

Energy level statistics of Hermitian random matrices $\hat H$ with Gaussian independent random entries $H_{i\geq j}$ is studied for a generic ensemble of almost diagonal random matrices with $ <|H_{ii}|^{2} > \sim 1$ and $<|H_{i\neq j}|^{2} >= b {\cal F}(|i-j|) \ll 1$. We perform a regular expansion of the spectral form-factor $K(\tau) = 1 + b K_{1}(\tau) + b^{2} K_{2}(\tau) + ... $ in powers of $b \ll 1$ with the coefficients $K_{m}(\tau)$ that take into account interaction of (m+1) energy levels. To calculate $K_{m}(\tau)$, we develop a diagrammatic technique which is based on the Trotter formula and on the combinatorial problem of graph edges coloring with (m+1) colors. Expressions for $K_{1}(\tau)$ and $K_{2}(\tau)$ in terms of infinite series are found for a generic function ${\cal F}(|i-j|)$ in the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE) and in the crossover between them (the almost unitary Gaussian ensemble). The Rosenzweig-Porter and power-law banded matrix ensembles are considered as examples.

Moments of vicious walkers and Möbius graph expansions.

A system of Brownian motions in one dimension all started from the origin and conditioned never to collide with each other in a given finite time interval (0,T] is studied. The spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue statistics of random Hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian orthogonal ensemble (GOE) statistics as the time t goes on from 0 to T. In the present paper, we characterize this GUE-to-GOE transition by presenting the graphical expansion formula for the moments of positions of vicious walkers. In the GUE limit t-->0, only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus. Following the time evolution of the vicious walkers, however, the graphs with twisted ribbons, called Möbius graphs, increase their contribution to our expansion formula, and we have to deal with the topology of nonorientable surfaces. Application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the Möbius expansion using the Stirling numbers of the first kind.

The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities

We discuss the CLT for the global and local linear statistics of random matrices from classical compact groups. The mainparts of our proofs are certain combinatorial identities, much in the spirit of works by M. Kac and H. Spohn.

Fluctuation formula for complex random matrices

A Gaussian fluctuation formula is proved for linear statistics of complex random matrices in the case that the statistic is rotationally invariant. For a general linear statistic without this symmetry, Coulomb gas theory is used to predict that the distribution will again be a Gaussian, with a specific mean and variance. The variance splits naturally into a bulk and surface contibution, the latter resulting from the long range correlations at the boundary of the support of the eigenvalue density.

On the linear statistics of Hermitian random matrices

In this paper we continue with the study of linear statistics of random Hermitian matrix ensembles. We generalize the result of a previous paper on the probability density function of linear statistics to the finite N ensembles where the interval of support of the eigenvalue spectrum is a single interval. Combining the Hankel determinant formula for orthogonal polynomials which are associated with Hermitian matrix ensembles and the linear statistics theorem for finite N, we obtain the strong or oscillatory asymptotics for the polynomials orthogonal with respect to weight functions supported on the real axis.

Logarithmic terms in the spectral statistics of band random matrices

The authors present a detailed analysis of the nearest-neighbour spacing distribution (NNSD) for random matrices on which the non-zero elements are confined to a band around the diagonal. The vanishing of the remaining elements induces non-analytic (logarithmic) behaviour for the NNSD at very small spacings. A conjective for the precise behaviour of these logarithmic terms is presented and supported by extensive numerical computations.

Eigenvector statistics of random band matrices.

Eigenvector statistics of random band matrices.