Asymptotic corrections to the Wigner semicircular eigenvalue spectrum of a large real symmetric random matrix using the replica method

Abstract

The replica method has previously been used to calculate the semicircular averaged eigenvalue spectrum of the Gaussian orthogonal ensemble of real symmetric N x N random matrices in the limit where N + CC. In this paper we develop a perturbative scheme which, within this same replica framework, is used to calculate the corrections within this semicircular band of eigenvalues to order 1/ N and 1/ N2. Comparison is made between these results and previously published work by other authors on the corrections to order 1/ N. A new and straightforward self-consistency argument is presented and used to derive the shape of the averaged eigenvalue spectrum when N is large but finite and the scaling behaviour of this averaged eigenvalue spectrum near the band edges is demonstrated in a straightforward fashion. Some comments are made on the relation of our results to those of field theoretical calculations in zero dimensions.

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

“…For the GOE, the departure from the semicircle becomes noticeable only for N <6 [41,43]. By comparison with the figures supplied in DJ [45], it can be seen that the significant departure is due to ρ large N (∼200), this broadening and overlapping tend to the semicircle. One should not expect ρ(λ) to mimic correctly the AED of the RSSME when N < 6. Corrections to O(1/N 2 ) will be required for these low values of 1/N.…”

“…This term subsequently sustains Eqs. (34) and (38) for the 1/N correction to ρ(λ), whence going beyond the DJ analysis [45].…”

“…A similar procedure goes for evaluating K [Eq. (32)] (see also DJ [45]). Notice that there is no p variable in K, whence the p integral = 1.…”

“…For the GOE, there has been a number of papers that have addressed the problem of calculating the corrections to the Wigner semicircle which are of order 1/N [42][43][44]. Dhesi and Jones [45] (thereafter referred to as DJ) have provided a comprehensive set of results: DJ calculated the AED for the GOE to order 1/N 2 , and also the AED for finite N based on a self-consistency argument, when each element of the matrix is drawn randomly from a normal distribution. Note that these four papers rely on different methods; furthermore, the 1/N correction of Takano and Takano [42] differs from the others.…”

“…(1) as treated in DJ [45]] converges to the semicircle as N becomes large. We show that additional terms appear when the finite size is taken into account.…”