A note on the large random inner-product kernel matrices

In this note we consider the n×nn×n random matrices whose (i,j)(i,j)th entry is f(xiTxj), where xixi’s are i.i.d. random vectors in RNRN, and ff is a real-valued function. The empirical spectral distributions of these random inner-product kernel matrices are studied in two kinds of high-dimensional regimes: n/N→γ∈(0,∞)n/N→γ∈(0,∞) and n/N→0n/N→0 as both nn and NN go to infinity. We obtain the limiting spectral distributions for those matrices from different random vectors in RNRN including the points lplp-norm uniformly distributed over four manifolds. And we also show a result on isotropic and log-concave distributed random vectors, which confirms a conjecture by Do and Vu.