Random Matrix Approximation of Spectra of Integral Operators

Let H:L₂(S,\ S,P) \→ L₂(S,\ S,P) be a compact integral operator with a symmetric kernel h. Let Xᵢ,\\ i\∈\ , be independent S-valued random variables with common probability law P. Consider the n×n matrix ̃ₙ with entries n⁻¹h(Xᵢ, Xⱼ),\\ 1\≤ i,j\≤ n (this is the matrix of an empirical version of the operator H with P replaced by the empirical measure Pn), and let Hn denote the modification of \ Hₙ, obtained by deleting its diagonal. It is proved that the ₂ distance between the ordered spectrum of Hn 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 theorems for the difference between the ordered spectra of the operators Hn (or \ Hₙ ) and H are also obtained under somewhat stronger conditions. These results apply in particular to the kernels of certain functions H=\φ (L) of partial differential operators L (heat kernels, Green functions).