Semicircle law for random matrices of long-range percolation model
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S. Ayadi
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 = dσn,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 .
© de Gruyter 2009
Articles in the same Issue
- Semicircle law for random matrices of long-range percolation model
- The stochastic maximum principle in optimal control of degenerate diffusions with non-smooth coefficients
- The asymptotic behaviour of the maximum of a random sample subject to trends in location and scale
- Evolution process as an alternative to diffusion process and Black–Scholes Formula
- Occupation time problems for fractional Brownian motion and some other self-similar processes
- Properties of the distribution of the random variable defined by A2-continued fraction with independent elements
Articles in the same Issue
- Semicircle law for random matrices of long-range percolation model
- The stochastic maximum principle in optimal control of degenerate diffusions with non-smooth coefficients
- The asymptotic behaviour of the maximum of a random sample subject to trends in location and scale
- Evolution process as an alternative to diffusion process and Black–Scholes Formula
- Occupation time problems for fractional Brownian motion and some other self-similar processes
- Properties of the distribution of the random variable defined by A2-continued fraction with independent elements
