Averaged Eigenvalue Spectrum of Large Symmetric Random Matrix

Abstract

The averaged eigenvalue spectrum of a large symmetric random matrix, in which each element is an independent random variable with the Gaussian distribution, is calculated by using the diagram technique. Compared with the methods used by Edwards and Jones and by Mehta, the present method is very simple and can be used in other calculations.
Leading terms in N (the dimension of the matrix) and next leading terms are calculated exactly. It is shown that the term proportional to N gives the semi-circular law obtained by Edwards and Jones. Next leading terms, which is of the order of unity, give three δ-functions as well as the corrections to the semi-circular law. One of the three δ-functions is the same as that of Edwards and Jones and other two are located at the band edges of the semi-circular law. The physical meanings of these results are discussed.