Spectral singularities of structured random matrices
A talk in the Oberseminar Probability Theory and Mathematical Statistics series by
Torben Krüger from FAU Erlangen
| Abstract: | As the dimension of a random matrix tends to infinity, its empirical eigenvalue distribution often converges to a deterministic limiting measure. For Hermitian random matrices this measure is supported on the real line, while for non-Hermitian matrices it is supported on the complex plane. In the basic Hermitian case of a matrix with independent, identically distributed entries above the diagonal, the limiting spectral measure is the famous semicircle law. Its density exhibits two standard types of local behavior: it is positive in the bulk of the spectrum, while at the spectral edges it vanishes with square-root behavior. Correspondingly, on the local eigenvalue-spacing scale, the eigenvalues are governed by two universal limit distributions: the sine kernel statistics in the bulk and the Airy kernel at the edge. When the distribution of the matrix entries has additional structure, for instance when the entries have non-identical variances, the limiting spectral density may develop further singularities beyond the usual bulk and edge regimes. These singularities lead to new local scaling limits and new universal eigenvalue statistics. In this talk, we give an overview of recent progress on the classification of such spectral singularities and on the corresponding local universal laws. |