Faculty of Mathematics
Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics
stripes SFB701

Project A4

Asymptotics of spectral distributions


Principal Investigator(s) Other Investigators
Friedrich Götze
Holger Kösters
Gennadiy P. Chistyakov
Holger Sambale
Martin Venker

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.

Links:

Recent Preprints:

16049 On the local semicircular law for Wigner ensembles PDF | PS.GZ
16048 Second order concentration via logarithmic Sobolev inequalities PDF | PS.GZ
16047 Rényi divergence and the central limit theorem PDF | PS.GZ
16045 Products of random matrices from polynomial ensembles PDF | PS.GZ
16044 Exact relation between singular value and eigenvalue statistics PDF | PS.GZ
16006 Limit theorems for number of edges in the generalized random graphs with random vertex weights PDF | PS.GZ
15106 Edge Statistics for a Class of Repulsive Particle Systems PDF | PS.GZ
15105 Empirical Spacings of Unfolded Eigenvalues PDF | PS.GZ
15104 Limiting Spectral Distributions of Sums of Products of non-Hermitian Random Matrices PDF | PS.GZ
15087 Asymptotic Expansions in Free Limit Theorems PDF | PS.GZ