Asymptotic Properties of Random Matrices of Long-Range Percolation Model

TL;DR

This paper investigates the spectral properties of long-range percolation matrices, deriving explicit correlation functions and establishing their universality class with band random matrices in the asymptotic limit.

Contribution

It provides explicit expressions for the correlation functions of long-range percolation matrices and demonstrates their spectral universality with band random matrices.

Findings

Derived explicit correlation function T(z1,z2) for the model.

Showed the eigenvalue density correlation matches band random matrices.

Established the universality class for the spectral properties.

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.