Asymptotics of spectral distributions
The main focus of this project is the investigation of asymptotic distributions of eigenvalues and eigenvectors of matrices of high dimensional random matrix ensembles as well as spectral distributions of infinite dimensional operators in free probability theory. Another focus is the connection of matrix-valued stochastic processes and their induced spectral processes to representation theory and the limits of related combinatorial structures like Young diagrams and partitions. Furthermore, representation theoretic methods will be used to compute asymptotic approximations to higher correlations of characteristic polynomials. The limiting local and global distributions of eigenvalues appearing in this context are often universal and appear as limiting objects in various contexts of mathematics as well as mathematical physics. A rather incomplete list contains representation theory, asymptotic combinatorics, nuclear growth models in probability, free probability and operator algebras, determinantal point processes, integrable systems as well as the correlations of zeros of L-functions. These similarities ask for an explanation in a more general framework. In this project, we intend to concentrate on some of these connections, connecting probability, algebraic combinatorics and complex analysis. In cooperation with a number of other projects of the CRC we hope to advance the understanding of these surprising connections between different fields.