Central Limit Theorem for Fluctuations of Linear Eigenvalue Statistics of Large Random Graphs. Dilut (original) (raw)
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We study the linear eigenvalue statistics of large random graphs in the regimes when the mean number of edges for each vertex tends to infinity. We prove that for a rather wide class of test functions the fluctuations of linear eigenvalue statistics converges in distribution to a Gaussian random variable with zero mean and variance which coincides with "non gaussian" part of the Wigner ensemble variance.
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Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
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For k, m, n ∈ N, we consider n k × n k random matrices of the form M n,m,k (y) = m α=1 τ α Y α Y T α , Y α = y (1) α ⊗ • • • ⊗ y (k) α , where τ α , α ∈ [m], are real numbers and y (j) α , α ∈ [m], j ∈ [k], are i.i.d. copies of a normalized isotropic random vector y ∈ R n. For every fixed k ≥ 1, if the Normalized Counting Measures of {τ α } α converge weakly as m, n → ∞, m/n k → c ∈ [0, ∞) and y is a good vector in the sense of Definition 1.1, then the Normalized Counting Measures of eigenvalues of M n,m,k (y) converge weakly in probability to a nonrandom limit found in Marchenko and Pastur (Math USSR Sb 1:457-483, 1967). For k = 2, we define a subclass of good vectors y for which the centered linear eigenvalue statistics n −1/2 Tr ϕ(M n,m,2 (y)) • converge in distribution to a Gaussian random variable, i.e., the Central Limit Theorem is valid.
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We study the variance and the characteristic functional of linear eigenvalue statistics of unitary invariant matrix models of n◊n Hermitian matrices as n ! 1. Assuming that the test function of the statistic is smooth enough and using the asymptotic formulas by Deift et al for orthogonal polynomials with varying weight, we show first that if the support of the density of states of the model consists of q 2 intervals, then in the global regime the variance is a quasi-periodic function of n as n ! 1 generically in the potential, determining the model. We show next that the exponent of the characteristic functional in general is not 1/2◊ the variance, as it should be if the Central Limit Theorem would be valid, and we find the asymptotic form of the characteristic functional in certain cases. We also show that in the intermediate and the local regimes the Central Limit Theorem is valid.
Spectral statistics of Erdős–Rényi graphs I: Local semicircle law
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We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, that is, graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as pN → ∞ (with a speed at least logarithmic in N), the density of eigenvalues of the Erdős-Rényi ensemble is given by the Wigner semicircle law for spectral windows of length larger than N −1 (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the ∞-norms of the 2-normalized eigenvectors are at most of order N −1/2 with a very high probability. The estimates in this paper will be used in the companion paper [Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues (2011) Preprint] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pN N 2/3 .
Annealed limit theorems for the Ising model on random regular graphs
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In a recent paper, Giardinà et al. [ALEA Lat. Am. J. Probab. Math. Stat. 13 (2016) 121-161] have proved a law of large number and a central limit theorem with respect to the annealed measure for the magnetization of the Ising model on some random graphs, including the random 2-regular graph. In this paper, we present a new proof of their results which applies to all random regular graphs. In addition, we prove the existence of annealed pressure in the case of configuration model random graphs.
Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues
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We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption pN ≫ N 2/3 , we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erdős-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments.
Asymptotic Normality of Some Graph-Related Statistics
Journal of Applied Probability, 1989
Petrovskaya and Leontovich (1982) proved a central limit theorem for sums of dependent random variables indexed by a graph. We apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. We briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics.