Real symmetric random matrices and path counting.

Exact evaluation of (TrS(p)) is here performed for real symmetric matrices S of arbitrary order n , up to some integer p , where the matrix entries are independent identically distributed random variables, with an arbitrary probability distribution. These expectations are polynomials in the moments of the matrix entries; they provide useful information on the spectral density of the ensemble in the large n limit. They also are a straightforward tool to examine a variety of rescalings of the entries in the large n limit.