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

Abstract

The recent paper "Finite size effects in the averaged eigenvalue density of Wigner random-sign real symmetric matrices" by G. S. Dhesi and M. Ausloos [Phys. Rev. E 93, 062115 (2016)10.1103/PhysRevE.93.062115] uses the replica method to compute the 1/N correction to the Wigner semicircle law for the ensemble of real symmetric random matrices with 0's down the diagonal, and upper triangular entries independently chosen from the values ±w with equal probability. We point out that the results obtained are inconsistent with known results in the literature, as well as with known large N series expansions for the trace of powers of these random matrices. An incorrect assumption relating to the role of the diagonal terms at order 1/N appears to be the cause for the inconsistency. Moreover, results already in the literature can be used to deduce the 1/N correction to the Wigner semicircle law for real symmetric random matrices with entries drawn independently from distributions D_{1} (diagonal entries) and D_{2} (upper triangular entries) assumed to be even and have finite moments. Large N expansions for the trace of the 2kth power (k=1,2,3) for these matrices can be computed and used as checks.