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.
Categories: math-ph math.MP math.PR