Asymptotic Properties of Random Matrices of Long-Range Percolation Model

We study the spectral properties of matrices of long-range percolation model. These are N\times N 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 with \psi(t)\le{1} and vanishing at infinity. We study the resolvent G(z)=(H-z)^{-1}, Imz\neq{0} in the limit N,b\to\infty, b=O(N^{\alpha}), 1/3