A note on the large random inner-product kernel matrices

In this note we consider the n×n random matrices whose (i,j)th entry is f(xiTxj), where xi’s are i.i.d. random vectors in RN, and f 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,∞) and n/N→0 as both n and N go to infinity. We obtain the limiting spectral distributions for those matrices from different random vectors in RN including the points lp-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.