25 research outputs found

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

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    We consider n×nn\times n real symmetric and Hermitian Wigner random matrices n1/2Wn^{-1/2}W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n1XXn^{-1}X^*X with independent entries of m×nm\times n matrix XX. Assuming first that the 4th cumulant (excess) κ4\kappa_4 of entries of WW and XX 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 nn\to\infty, mm\to\infty, m/nc[0,)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 C5\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 C5\mathbb{C}^5 test function. Here the variance of statistics contains an additional term proportional to κ4\kappa_4. The proofs of all limit theorems follow essentially the same scheme.Comment: Published in at http://dx.doi.org/10.1214/09-AOP452 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    On Non-Gaussian Limiting Laws for Certain Statistics of Wigner Matrices

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    This paper is a continuation of our papers [12-14] in which the limiting laws of fluctuations were found for the linear eigenvalue statistics Tr φ(M⁽ⁿ⁾) and for the normalized matrix elements √nφjj(M⁽ⁿ⁾) of differentiable functions of real symmetric Wigner matrices M⁽ⁿ⁾ as n →∞.Статья является продолжением исследования, начатого в работах [12-14],где были найдены предельные законы флуктуаций для линейных статистик собственных значений Tr φ(M⁽ⁿ⁾) и нормированных матричных элементов √nφjj(M⁽ⁿ⁾) дифференцируемых функций от вещественных симметричных матриц Вигнера M⁽ⁿ⁾ при n →∞

    Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices

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    We study the fluctuations of the matrix entries of regular functions of Wigner random matrices in the limit when the matrix size goes to infinity. In the case of the Gaussian ensembles (GOE and GUE) this problem was considered by A.Lytova and L.Pastur in J. Stat. Phys., v.134, 147-159 (2009). Our results are valid provided the off-diagonal matrix entries have finite fourth moment, the diagonal matrix entries have finite second moment, and the test functions have four continuous derivatives in a neighborhood of the support of the Wigner semicircle law.Comment: minor corrections; the manuscript will appear in the Journal of Statistical Physic

    On universality of local edge regime for the deformed Gaussian Unitary Ensemble

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    We consider the deformed Gaussian ensemble Hn=Hn(0)+MnH_n=H_n^{(0)}+M_n in which Hn(0)H_n^{(0)} is a hermitian matrix (possibly random) and MnM_n is the Gaussian unitary random matrix (GUE) independent of Hn(0)H_n^{(0)}. Assuming that the Normalized Counting Measure of Hn(0)H_n^{(0)} converges weakly (in probability if random) to a non-random measure N(0)N^{(0)} with a bounded support and assuming some conditions on the convergence rate, we prove universality of the local eigenvalue statistics near the edge of the limiting spectrum of HnH_n.Comment: 25 pages, 2 figure

    Functional limit theorems for random regular graphs

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    Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as n grows to infinity, either when d is kept fixed or grows slowly with n. In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein's method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn-Szemer\'edi argument for estimating the second largest eigenvalue for all values of d and n.Comment: Added Remark 27. 39 pages. To appear in Probability Theory and Related Field