The Eigenvalues of Very Sparse Random Symmetric Matrices

Let W_n be an n × n random symmetric sparse matrix with independent identically distributed entries such that the values 1 and 0 are taken with probabilities p/n and 1-p/n, respectively; here is independent of n. We show that the limit of the expected spectral distribution functions of W_n has a discrete part. Moreover, the set of positive probability points is dense in (- +). In particular, the points , and 0 belong to this set.