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Semicircle law for random matrices of long-range percolation model

Abstract We study the normalized eigenvalue counting measure dσ of matrices of long-range percolation model. These are (2n + 1) × (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 ∈ ℝ+, where ψ is an even positive function ψ(t) ≤ 1 vanishing at infinity. It is shown that if the third moment of , is uniformly bounded, then the measure dσ = dσ n,b weakly converges in probability in the limit n, b → ∞, 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 gn,b (z) converges in average and that the variance of gn,b (z) vanishes. In the second part of the paper, we estimate the rate of decreasing of the variance of gn,b (z) under further conditions on the moments of .