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

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)] 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 $\pm w$ 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 semicircle 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 $2k\mathrm{th}$ power $\left(k=1,2,3\right)$ for these matrices can be computed and used as checks.