On the asymptotic spectral distribution of increasing size matrices: test functions, spectral clustering, and asymptotic estimates of outliers

Let Ann be a sequence of square matrices such that size (An) =dn→∞ as n→∞. We say that Ann has an asymptotic spectral distribution described by a Lebesgue measurable function f: D⊂Rk→C if, for every continuous function F: C→C with bounded support, limn→∞⁡1dn∑i=1dnF (λi (An) ) =1μk (D) ∫DF (f (x) ) dx, where μk is the Lebesgue measure in Rk and λ1 (An), …, λdn (An) are the eigenvalues of An. In the last decades, the asymptotic spectral distribution of increasing size matrices has become a subject of investigation by several authors. Special attention has been devoted to matrices An arising from the discretization of differential equations, whose size dn diverges to ∞ along with the mesh-fineness parameter n. However, despite the popularity that the topic has reached nowadays, the role of the so-called test functions F appearing in the above limit relation has been inexplicably neglected so far. In particular, a natural question such as "Which is the largest set of test functions F for which the above limit relation is satisfied? " is still unanswered. In the present paper, we provide a definitive answer to this question by identifying the largest set of test functions F for which the above limit relation is satisfied. We also present some applications of this result to the analysis of spectral clustering, including new asymptotic estimates on the number of outliers. A special attention is devoted to the so-called "inner outliers" of Hermitian block Toeplitz matrices. We conclude the paper with an interpretation of the main result in the context of the vague convergence of probability measures.