Semicircle Law for Random Matrices of Long-Range Percolation Model

Abstract

We study the normalized eigenvalue counting measure d\sigma of matrices of long-range percolation model. These are (2n+1)\times (2n+1) random real symmetric matrices H=\{H(i,j)\}_{i,j} whose elements are independent random variables taking zero value with probability 1-\psi [(i-j)/b], b\in \mathbb{R}^{+}, where \psi is an even positive function \psi(t)\le{1} vanishing at infinity. It is shown that if the third moment of \sqrt{b}H(i,j), i\leq{j} is uniformly bounded then the measure d\sigma:=d\sigma_{n,b} weakly converges in probability in the limit n,b\to\infty, b=o(n) to the semicircle (or Wigner) distribution. The proof uses the resolvent technique combined with the cumulant expansions method. We show that the normalized trace of resolvent g_{n,b}(z) converges in average and that the variance of g_{n,b}(z) vanishes. In the second part of the paper, we estimate the rate of decreasing of the variance of g_{n,b}(z), under further conditions on the moments of \sqrt{b}H(i,j), \ i\le{j}.