Semicircle Law for Random Matrices of Long-Range Percolation Model

TL;DR

This paper proves that the eigenvalue distribution of large long-range percolation matrices converges to the semicircle law, using resolvent and cumulant techniques, under certain moment conditions.

Contribution

It establishes the semicircle law for a broad class of long-range percolation matrices and provides convergence rates under moment assumptions.

Findings

Eigenvalue measure converges to semicircle distribution as matrix size grows.

Normalized resolvent trace converges in average, variance vanishes.

Provides estimates on the rate of variance decrease under moment conditions.

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}.