High Moments of Large Wigner Random Matrices and Asymptotic Properties of the Spectral Norm

We consider an ensemble of nxn real symmetric random matrices A whose entries are determined by independent identically distributed random variables that have symmetric probability distribution. Assuming that the moment 12+2delta of these random variables exists, we prove that the probability distribution of the spectral norm of A rescaled to n^{-2/3} is bounded by a universal expression. The proof is based on the completed and modified version of the approach proposed and developed by Ya. Sinai and A. Soshnikov to study high moments of Wigner random matrices.Comment: This version: misprints corrected, some parts of the proofs simplified, general presentation improved. The final version to appear in: Random Operators and Stoch. Equation

A class of even walks and divergence of high moments of large Wigner random matrices

We study high moments of truncated Wigner nxn random matrices by using their representation as the sums over the set W of weighted even closed walks. We construct the subset W' of W such that the corresponding sum diverges in the limit of large n and the number of moments proportional to n^{2/3} for any truncation of the order n^{1/6+epsilon}, epsilon>0 provided the probability distribution of the matrix elements is such that its twelfth moment does not exist. This allows us to put forward a hypothesis that the finiteness of the twelfth moment represents the necessary condition for the universal upper bound of the high moments of large Wigner random matrices.Comment: version 2: minor changes; formulas (1.4) and (2.2) correcte

On eigenvalue distribution of random matrices of Ihara zeta function of large random graphs

We consider the ensemble of real symmetric random matrices $H^{(n,\rho)}$ obtained from the determinant form of the Ihara zeta function of random graphs that have $n$ vertices with the edge probability $\rho/n$. We prove that the normalized eigenvalue counting function of $H^{(n,\rho)}$ weakly converges in average as $n,\rho\to\infty$ and $\rho=o(n^\alpha)$ for any $\alpha>0$ to a shift of the Wigner semi-circle distribution. Our results support a conjecture that the large Erdos-R\'enyi random graphs satisfy in average the weak graph theory Riemann Hypothesis.Comment: version 5: slightly corrected with respect to version

On asymptotic properties of high moments of compound Poisson distribution

We study asymptotic properties of the moments $M_k(\lambda)$ of compound Poisson random variable $\sum_{j=1}^N W_j$. These moments can be regarded as a weighted version of Bell polynomials. We consider a limiting transition when the number of the moment $k$ infinitely increases at the same time as the mean value $\lambda$ of the Poisson random variable $N$ tends to infinity and find an explicit expression for the rate function in dependence of the ratio $k/\lambda$. This rate function is expressed in terms of the exponential generating function of the moments of $W_j$. We illustrate the general theorem by three particular cases corresponding to the normal, gamma and Bernoulli distributions of the weights $W_j$. We apply our results to the study of the concentration properties of weighted vertex degree of large random graphs. Finally, as a by-product of the main theorem, we determine an asymptotic behavior of the number of even partitions in comparison with the asymptotic properties of the Bell numbers.Comment: 31 page

Estimates for moments of random matrices with Gaussian elements

We describe an elementary method to get non-asymptotic estimates for the moments of Hermitian random matrices whose elements are Gaussian independent random variables. As the basic example, we consider the GUE matrices. Immediate applications include GOE and the ensemble of Gaussian anti-symmetric Hermitian matrices. The estimates we derive give asymptotically exact expressions for the first terms of 1/N-expansions of the moments and covariance terms. We apply our method to the ensemble of Gaussian Hermitian random band matrices whose elements are zero outside of the band of width b. The estimates we obtain show that the spectral norm of these matrices remains bounded in the limit of infinite N when b is much greater than log N to the power 3/2.Comment: 45 pages (the version improved after the referee reports; one diagram added

On Eigenvalue Distribution of Random Matrices of Ihara Zeta Function of Large Random Graphs

We consider the ensemble of real symmetric random matrices H(n,ρ) obtained from the determinant form of the Ihara zeta function of random graphs that have n vertices with the edge probability ρ/n. We prove that the normalized eigenvalue counting function of H(n,ρ) converges weakly in average as n, ρ→∞ and ρ = o(nα) for any α > 0 to a shift of the Wigner semi-circle distribution. Our results support a conjecture that the large Erdős-Rényi random graphs satisfy in average the weak graph theory Riemann Hypothesis

On high moments of strongly diluted large Wigner random matrices

We consider a dilute version of the Wigner ensemble of nxn random matrices $H$ and study the asymptotic behavior of their moments $M_{2s}$ in the limit of infinite $n$, $s$ and $\rho$, where $\rho$ is the dilution parameter. We show that in the asymptotic regime of the strong dilution, the moments $M_{2s}$ with $s=\chi\rho$ depend on the second and the fourth moments of the random entries $H_{ij}$ and do not depend on other even moments of $H_{ij}$. This fact can be regarded as an evidence of a new type of the universal behavior of the local eigenvalue distribution of strongly dilute random matrices at the border of the limiting spectrum. As a by-product of the proof, we describe a new kind of Catalan-type numbers related with the tree-type walks.Comment: 43 pages (version four: misprints corrected, discussion added, other minor modifications