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

The replica method has previously been used to calculate the semicircular averaged eigenvalue spectrum of the Gaussian orthogonal ensemble of real symmetric N*N random matrices in the limit where N to infinity . The authors 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. 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 the results to those of field theoretical calculations in zero dimensions.