A note on the large random inner-product kernel matrices

In this note we consider the n x n random matrices whose (i, j)th entry is f(x(i)(T)x(j)), where x(i)'s are i.i.d. random vectors in R-N, 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 -> gamma is an element of (0, infinity) 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 R-N including the points l(p)-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. (C) 2015 Published by Elsevier B.V.