Random matrix approximation of spectra of integral operators

Let H:L-2(S, L, P) bar right arrow L-2(S, L, P) be a compact integral opera tor with a symmetric kernel h. Let X-i, i is an element of N, be independen t S-valued random variables with common probability law P. Consider the n X n matrix (H) over tilde(n) with entries n(-1) h(X-i, X-j), 1 less than or equal to i, j less than or equal to n (this is the matrix of an empirical v ersion of the operator H with P replaced by the empirical measure P-n), and let H-n denote the modification of (H) over tilde(n), obtained by deleting its diagonal. It is proved that the l(2) distance between the ordered spec trum of H-n and the ordered spectrum of H tends to zero a.s. if and only if H is Hilbert-Schmidt. Rates of convergence and distributional limit theore ms for the difference between the ordered spectra of the operators H-n (or (H) over tilde(n)) and H are also obtained under somewhat stronger conditio ns. These results apply in particular to the kernels of certain functions H = phi(L) of partial differential operators L (heat kernels, Green function s).