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 M2s(nρ) = ETr(H(n,ρ)2s in the limit when n, s and ρ tend to infinity. Our main result is that the sequence M2sn(n,ρn) with sn = ⌊χρn⌋, χ > 0 and ρn = o(n1/5) is asymptotically close to a sequence of numbers nmsn(ρn), where {ms(ρ)}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, S ∈ N. It shows that the spectral properties of random matrices at the edge of the limiting spectrum in the asymptotic regime of the strong dilution essentially differ from those observed in the case of the weak dilution, where the dependence on the fourth moment V4 does not intervene.