Semicircle Law for Random Matrices of Long-Range Percolation Model
Abstract
Description
We study the normalized eigenvalue counting measure dσof 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-ψ[(i-j)/b], b\in \mathbb{R}^{+}, where ψis an even positive function ψ(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 dσ:=dσ_{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}.
The author is grateful to Prof. Dr. O. Khorunzhy at University of Versailles (France), where present paper was completed, who proposed use to study the problems described in this paper
The author is grateful to Prof. Dr. O. Khorunzhy at University of Versailles (France), where present paper was completed, who proposed use to study the problems described in this paper