Some Remarks on the Eigenvalue Spectrum of A Large Symmetrical Wigner Random-Sign Matrix

The replica method is used to calculate the averaged eigenvalue spectrum as N --> infinity, of the ensemble of Wigner random-sign real symmetric N X N matrices. Results are presented for the cases where the individual matrix elements have a mean value of zero and also where the mean value of the individual matrix elements has a finite nonzero value. It is shown that the replica method provides a straightforward framework within which it is possible to verify the Wigner conjecture that any reasonably well-behaved distribution of matrix elements must lead to the well-known semicircular averaged eigenvalue spectrum of the Gaussian orthogonal ensemble of random matrices. Some numerical simulations of the averaged eigenvalue spectrum of these random-sign matrices are presented and they lend support to the prediction that if the individual matrix elements have sufficiently large a mean value, then a single eigenvalue will split off from the main semicircular band of eigenvalues.