Finite size effects. The averaged eigenvalue density of Wigner random sign real symmetric matrices

TL;DR

This paper analytically derives the first finite size correction to the eigenvalue density and Green function of Wigner random sign real symmetric matrices using the replica method, validated by simulations.

Contribution

It introduces a perturbative scheme within the replica framework to obtain finite size corrections for eigenvalue spectra of Wigner matrices, a novel analytical approach.

Findings

Analytical expressions for eigenvalue density to order 1/N

Excellent agreement between theory and numerical simulations

First finite size correction derived and validated

Abstract

Nowadays, strict finite size effects must be taken into account in condensed matter problems when treated through models based on lattices or graphs. On the other hand, the cases of directed bonds or links are known as highly relevant, in topics ranging from ferroelectrics to quotation networks. Combining these two points leads to examine finite size random matrices. To obtain basic materials properties, the Green function associated to the matrix has to be calculated. In order to obtain the first finite size correction a perturbative scheme is hereby developed within the framework of the replica method. The averaged eigenvalue spectrum and the corresponding Green function of Wigner random sign real symmetric N x N matrices to order 1/N are in fine obtained analytically. Related simulation results are also presented. The comparison between the analytical formulae and finite size matrices numerical diagonalization results exhibits an excellent agreement, confirming the correctness of the first order finite size expression.