Comment on "Finite size effects in the averaged eigenvalue density of Wigner random-sign real symmetric matrices" by G.S. Dhesi and M. Ausloos

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 (2016), 062115] uses the replica method to compute the \(1/N\) correction to the Wigner semi-circle law for the ensemble of real symmetric random matrices with \(0\)'s down the diagonal, and upper triangular entries independently chosen from the values \(\pm w\) 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 semi-circle law for real symmetric random matrices with entries drawn independently from distributions \(\mathcal D_1\) (diagonal entries) and \(\mathcal D_2\) (upper triangular entries) assumed to be even and have finite moments. Large \(N\) expansions for the trace of the \(2k\)-th power (\(k=1,2,3\)) for these matrices can be computed and used as checks.