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

TL;DR

This paper critiques a recent study on finite size effects in Wigner matrices, highlighting inconsistencies with established results and correcting the approach to calculating 1/N corrections.

Contribution

It identifies errors in the previous work's assumptions and demonstrates how existing literature can be used to accurately compute 1/N corrections for Wigner matrices.

Findings

The previous results are inconsistent with known large N series expansions.

Corrected 1/N corrections can be derived using established large N expansions.

Existing literature provides reliable methods for calculating moments of Wigner 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 (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.