Distributions for the eigenvalues of large random matrices generated from four manifolds

In this paper we will first study the spectrum of certain large Euclidean random matrices. The entries of thesematrices are functions of n random points l(p)-norm uniformly sampled in N dimensional l(p) ball or l(p) sphere. Under the setting n/N -> 0, as N and n tend to infinity, it is shown that the empirical distributions of eigenvalues of normalized Euclidean random matrices generated from the above two manifolds both converge weakly to semicircle law. We also investigate certain 'sample-covariance' type matrices whose entries are determined by n random positions from an N dimensional l(p) ellipsoid or its surface. When N -> infinity and n -> infinity with N/n -> 0, we prove that their empirical spectral distribution converge to the same fixed distribution.