Asymptotic properties of random matrices of long-range percolation model

Abstract

We study the spectral properties of matrices of long-range percolation model. These are N × N random real symmetric matrices H = {H(i, j)}_i,j whose elements are independent random variables taking zero value with probability , where ψ is an even positive function with ψ(t) ≤ 1 and vanishing at infinity. We study the resolvent G(z) = (H – z)^–1, Im z ≠ 0, in the limit N, b → ∞, b = O(N^α), 1/3 < α < 1, and obtain the explicit expression T(z₁, z₂) for the leading term of the correlation function of the normalized trace of the resolvent g_N,b(z) = N^–1TrG(z). We show that in the scaling limit of local correlations, this term leads to the expression found earlier by other authors for band random matrix ensembles. This shows that the ratio b²/N is the correct scale for the eigenvalue density correlation function and that the ensemble we study and that of band random matrices belong to the same class of spectral universality.