Markus Ebke

The Spectral Singularity of Structured Random Matrices

Random matrix theory is a useful tool in mathematical physics for the statistical description of highly complex systems. One of the best understood models is the Gaussian Unitary Ensemble (GUE), which consists of complex Hermitian matrices with independent and identically distributed Gaussian entries, up to Hermiticity. Although the GUE is exactly solvable, it only applies to systems with a high degree of symmetry. Much less is understood if the matrix entries are independent but no longer identically distributed, a setting that is needed to model inhomogeneous systems.

In this talk, I will discuss Hermitian Gaussian random matrices with a block structure, where each block has its own variance. It is known that when the variance profile is sparse (i.e. the random matrix contains many zero blocks), then the spectral density can develop a singularity at the origin as the matrix size goes to infinity. I will present recent work on how to compute the microscopic scaling limit of the density at the origin, using anti-commuting variables, the superbosonization formula and a saddle-point approximation.

Based on joint work with Torben Krüger (FAU Erlangen-Nuremberg).