Faculty of Mathematics
Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics

# Project A4

## Asymptotics of spectral distributions

Principal Investigator(s) Other Investigators

## Summary:

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.

 13029 Semicircle law for a class of random matrices with dependent entries PDF | PS.GZ 13023 Large cycles and a functional central limit theorem for generalized weighted random permutations PDF | PS.GZ 12151 The entropic Erdös-Kac limit theorem PDF | PS.GZ 12118 Particle systems with repulsion exponent $\beta$ and random matrices PDF | PS.GZ 12111 Meixner class of non-commutative generalized stochastic processes with freely independent values. I. A characterization PDF | PS.GZ 12064 The arithmetic of distributions in free probability theory PDF | PS.GZ 12062 Free Infinitely Divisible Approximations of n-fold Free Convolutions PDF | PS.GZ 12049 Local universality of repulsive particle systems and random matrices PDF | PS.GZ 12046 Geometric sensitivity of random matrix results: consequences for shrinkage estimators of covariance and related statistical methods PDF | PS.GZ 12044 Elliptic law for real random matrices PDF | PS.GZ