Universal scaling limits at the spectral singularity of structured random matrices
A talk in the Seminar Zufallsmatrizen series by
Markus Ebke
| Abstract: | This talk focuses on the spectral properties of Hermitian K×K-block random matrices with independent centred entries and block-dependent variances. Such matrices are useful for modelling inhomogeneous systems, e.g. clustered random graphs with different coupling strengths within and between K clusters. It is known that when the variance profile is sparse (i.e. the random matrix contains many zero blocks), the spectral density develops a singularity at the origin as the dimension of the blocks goes to infinity. We compute the microscopic scaling limit of the density at the origin. For a low number of blocks (K=2 and K=3) we find that it depends only on the pattern of zero blocks but not on the specific values of the variances. A complete classification of possible limits for arbitrary K is in progress. Our derivation is based on an exact integral expression for the Stieltjes-transform of the density that we obtain using anti-commuting variables and the superbosonization formula. The scaling limit then follows via a saddle-point approximation. Based on joint work with Torben Krüger (arXiv:2511.19308). Within the CRC this talk is associated to the project(s): C6 |