Special Seminar in Computing and Mathematical Sciences

We consider n-by-n matrices whose (i, j)-th entry is f(X_i^T X_j), where X₁, ...,Xn are i.i.d. standard Gaussian random vectors in R^p, and f is a real-valued function. The eigenvalue distribution of these kernel random matrices is studied in the "large p, large n" regime. It is shown that with suitable normalization the spectral density converges weakly, and we identify the limit. Our analysis applies as long as the rescaled kernel function is generic, and particularly, this includes non-smooth functions, e.g. Heaviside step function. The limiting densities "interpolate" between the Marcenko-Pastur density and the semi-circle density.