Universität Bielefeld

Non-Hermitian random matrices with complex eigenvalues represent a truly two-dimensional (2D) Coulomb gas at inverse temperature beta=2. Compared to their Hermitian counter-parts they enjoy an enlarged bulk and edge universality. As an application to ecology we model large scale data of the approximately 2D distribution of buzzard nests in the Teutoburger forest observed over a period of 20 y. These birds of prey show a highly territorial behaviour. Their occupied nests are monitored annually and we compare these data with a one-component 2D Coulomb gas of repelling charges as a function of beta. The nearest neighbour spacing distribution of the nests is well described by fitting to beta as an effective repulsion parameter, that lies between the universal predictions of Poisson (beta=0) and random matrix statistics (beta=2). Using a time moving average and comparing with next-to-nearest neighbours we examine the effect of a population increase on beta and correlation length.
 

We report on recent developments in the study of nonlocal operators. The central object of the talk are quadratic forms similar to those that define Sobolev spaces of fractional order. These objects are naturally linked to Markov processes via the theory of Dirichlet forms. We provide regularity results for solutions to corresponding integrodifferential equations. Our emphasis is on forms with singularand anisotropic measures. Some of the objects under consideration are related to the Boltzmann equation, which leads to an interesting question of comparability of quadrativ forms. The talk is based on recent results joint with B. Dyda and with K.-U. Bux and T. Schulze.