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

Abstract

In this paper we will first study the spectrum of certain large Euclidean random matrices. The entries of these matrices are functions of n random points lp-norm uniformly sampled in N dimensional lp ball or lp sphere. Under the setting n/N → 0, as N and n tend to ∞, 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 lp ellipsoid or its surface. When N → ∞ and n → ∞ with N/n → 0, we prove that their empirical spectral distribution converge to the same fixed distribution.