Goals of Research Area G

Random matrices.

An important goal in random matrix theory is the extension of the universality of local eigenvalue correlations to more structured ensembles. For example, band matrices, whose entries are independent random variables with variance decaying rapidly to zero outside a band of size W, are expected to exhibit the same local correlations as Wigner matrices. Band matrices are particularly interesting because they interpolate between Wigner matrices and random Schrödinger operators where the disorder is restricted on the diagonal of the matrix (and the off-diagonal components, describing for example a discrete Laplacian, are deterministic). One can hope, therefore, that the study of band matrices will lead, in the long run, to a better mathematical understanding of random Schrödinger operators, which play an important role in condensed matter physics. In connection with growth-models and exclusion processes, there remains a challenging conjecture of the universality of the KPZ class in models that are not exactly solvable.

Optimal transport.

A central topic of future research will be optimal transport between random measures. The starting point of this major program is a paper under review by Sturm and his BIGS-student Huesmann where they construct an optimal coupling of the Lebesgue measure on Rd and the Poisson point process. It establishes the missing link between recent developments in optimal transport and allocation problems and also relates to current research of Karpinski (Research Area L) and Moldovanu (Research Area I). Another main topic will be optimal transportation on discrete spaces, with focus on the concept of gradient flows w.r.t. suitable ‘Wasserstein distances’ for Markov chains and jump processes. First results had already been obtained in the group around Sturm, mainly by long-term visitor Maas and BIGS-student Erbar. Moreover, the project of constructing and analyzing the Wasserstein diffusion on multi-dimensional spaces will be continued. From a more abstract perspective, this amounts to the study of stochastic evolutions on the diffeomorphism group (or its appropriate extensions). As a first step in this direction will be the detailed analysis of the entropic measure on multi-dimensional spaces constructed recently by Sturm.

Dynamics in random environments.

Up to now, virtually all results on the dynamics of spin glasses were limited to a spatial type of dynamics where the random environment affects only the holding times, but not the trace of the trajectory. The next challenge is to move beyond this limitation and to study, e.g. processes with Metropolis rates. This seems feasible using the coarse graining techniques. The progress on the extremal structure of BBM is opening the gate to push towards a much wider class of random processes with strong correlations, and the eventual hope is to reconnect to the study of ground states of spin glasses with non-hierarchical correlation structure. These threads are already picked up in several thesis projects.

Metastability.

Analyzing metastability and nucleation in stochastic Ising systems and lattice gases remains a key objective. Bovier and den Hollander, during his tenure as Bonn Research Chair in 2010, initiated the writing of a comprehensive monograph on this subject. A central point in the future development of the field will be the the analysis of metastability of spdes in higher spatial dimensions. It is well-known that spdes in dimension larger than one involve substantial new difficulties, and this is also the case for the analysis of their metastable behavior. This will involve interesting connections to analysis and we plan to intensify collaborations with S. Müller. On the applied side, triggered by a HCM workshop ‘Mathematical Biology’ organized in 2009, we have already started to investigate metastability effects in coagulation/fragmentation models inspired by the modeling of neurodegenerative diseases, and we intend to intensify this line of research, again making contact to research in Research Area B, in particular with Velasquez.

Stochastic algorithms and probabilistic models in economics.

Our study of equilibria in random dynamical games with interaction will be continued and extended. We aim at surpassing the Brownian motion paradigm of the contest games studied so far. We will take up BSDEs as a promising tool for characterizing equilibria in stochastic games in continuous time, to get new insights in the interdependence of the agents’ actions. A central future research topic are rigorous bounds for mixing times and non-asymptotic error estimates for Metropolis-Hastings algorithms in high dimensions and on function spaces. A prototypical application is transition path sampling, which is an MCMC method on path space. Although corresponding methods are applied in numerous areas, rigorous error estimates are still not available even in simple convex situations. The plan is to study carefully the convex Euclidean case, and to make first steps towards understanding more delicate models with lack of convexity or non-trivial geometry.