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

Abstract

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.

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“…Finite size effects are thus not artefacts, they are real effects which are automatically contained in our model. In the different context of random matrix theory for dynamical systems, there is recent progress in studying finite size effects [ 22 ], [ 23 ]. In our modeling, Eq (5) with the stationary covariance matrix Σ serves as a starting point.…”

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

“…Substituting ( 12), ( 13) in (5) with φ(λ) = 1 leads to a contradiction: the results (13) imply that we must have…”

“…In a recent work published in this journal by Dhesi and Ausloos [5], the spectral density of Wigner's original class of random matrices J N -now without the restriction to N odd -was reconsidered. In particular for N large but finite the leading correction term to the RHS of (1) was sought.…”

“…In particular for N large but finite the leading correction term to the RHS of (1) was sought. Earlier [6], the first author of [5] together with Jones had found this leading correction in the case of the Gaussian orthogonal ensemble (GOE) -real symmetric matrices with diagonal entries independent normal distributions N[0, √ 2w] and off-diagonal entries independent normal distributions N[0, w]; see e.g. [7, §1.1].…”

“…In our notation, we have factored the 1/N as seen in ( 8), whereas in [5] the 1/N factor is included in (10), (11). It is commented [5, below (57)] that the first of the terms in (8) proportional to 1/N corresponds to the 1/N correction to G(x) for the Gaussian orthogonal ensemble.…”