On High Moments of Strongly Diluted Large Wigner Random Matrices

Abstract

We consider a dilute version of the Wigner ensemble of n × n random real symmetric matrices H (n,ρ) , where ρ denotes the average number of non-zero elements per row. We study the asymptotic properties of the moments M (n,ρ) 2s = E Tr(H (n,ρ) ) 2s in the limit when n, s and ρ tend to infinity.Our main result is that the sequence M (n,ρn) 2snwith sn = ⌊χρn⌋, χ > 0, ρn → ∞ and ρn = o(n 1/5 ) is asymptotically close to a sequence of numbers nmsn , where {ms } s≥0 are determined by an explicit recurrence that involves the second and the fourth moments of the random variables (H (n,ρ) )ij , V2 and V4, respectively. This recurrent relation generalizes the one that determines the moments of the Wigner's semicircle law given by ms = limρ→∞ms(ρ), s ∈ N. It shows that the spectral properties of random matrices at the edge of the limiting spectrum regarded in the asymptotic regime of the strong dilution essentially differ from those observed in the cases of moderate and weak dilution, where the dependence on the fourth moment V4 does not intervene.

On the Correlation Functions of the Characteristic Polynomials of the Sparse Hermitian Random Matrices

“…The results of Theorem 3 are in a good agreement with the results of [12,13] in the sense that the asymptotic behavior changes when p crosses the rate n 2/3 . However, in [13] it is argued that in the case p n 2/3 the appropriate scale is p −1 instead of n −2/3 . We postpone the study of F 2 with the scaling p −1 , as well as the related study of F 2 near λ * ( p) for finite p, to subsequent publications.…”

“…First for p n 2/3 and then for p n ε with any ε > 0 it was shown that the spectral correlation functions of sparse hermitian random matrices in the bulk of the spectrum converge in the weak sense to that of GUE. For the edge of the spectrum, it was proved in [12] that for p n 2/3 the limiting probability P{max j λ (n) j > 2 + x/n 2/3 } admits a certain universal upper bound, whereas the result of [13] implies that for p n 1/5 the limiting probability P{max j λ (n) j > 2 + x/ p} is zero. Note that more advanced results for the edge eigenvalue statistics were obtained in [28] for so-called random d-regular graphs.…”

Limiting eigenvalue distribution of random matrices of Ihara zeta function of long-range percolation graphs

“…These interesting questions remain open ones that would require deep studies of fine spectral properties of the random matrix ensembles, such as the asymptotic behavior of the maximal eigenvalue of large random matrices A (R,φ) N (1.4). In these studies, additional restrictions could be imposed; these restrictions could be related with the ratio between N, R and φ 1 for the ensemble H (R) N (v) and between ρ and N for the random matrix of [16], where the value ρ = log N is shown to be critical for the asymptotic behavior of the spectral norm (see [15] for more details).…”