Asymptotic Properties of Random Matrices of Long-Range Percolation Model

Abstract

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<\alpha<1 and obtain the explicit expression T(z_{1},z_{2}) for the leading term of the correlation function of the normalized trace of resolvent g_{N,b}(z)=N^{-1}Tr G(z). We show that in the scaling limit of local correlations, this term leads to the expression (Nb)^{-1}T(\lambda+r_{1}/N+i0,\lambda+r_{2}/N-i0)= b^{-1}\sqrt{N}|r_{1}-r_{2}|^{-3/2}(1+o(1)) found earlier by other authors for band random matrix ensembles. This shows that the ratio $b^{2}/N$ is the correct scale for the eigenvalue density correlation function and that the ensemble we study and that of band random matrices belong to the same class of spectral universality.