Finite size effects in the averaged eigenvalue density of Wigner random-sign real symmetric matrices

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 to be highly relevant in topics ranging from ferroelectrics to quotation networks. Combining these two points leads us to examine finite size random matrices. To obtain basic materials properties, the Green's function associated with the matrix has to be calculated. 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's function of Wigner random sign real symmetric N×N matrices to order 1/N are finally obtained analytically. Related simulation results are also presented. The agreement is excellent between the analytical formulas and finite size matrix numerical diagonalization results, confirming the correctness of the first-order finite size expression.