# Achievements of Research Area G

## Random matrices.

Random matrix theory has links with many different branches of mathematics and physics. With the recent appointments of Rauhut (Bonn Junior Fellow), Ferrari, and Schlein, a very strong group working on different aspects of random matrices has been established. Erdös, Ramirez, Schlein, Tao, Vu, and Yau [G:EPR+10, G:ESY11, G:ERS+10] proved the longstanding universality conjecture for the eigenvalues statistics of Wigner matrices. Ferrari studies problems of random matrices in their connection to stochastic growth models and interacting particle systems. Outstanding contributions are the discovery of new universality classes for the limiting processes of growth models and exclusion processes, in collaborations with Adler and van Moerbeke [G:AFvM10] and Baik and Sandrine Peché [G:BFP10]. Rauhut (Bonn Junior Fellow) uses random matrix theory in compressed sensing, an emerging technique in signal processing. This topic creates an interesting and unexpected link with Research Area J.

## Optimal transport.

’Optimal transport’ has seen a decade of impressive, rapid developments. A major team of young international scientists in this emerging and innovative field of current mathematical research could be brought together in the group of Sturm, partly funded by HCM (Shao) or IAM (Gigli, laureate of the Oberwolfach prize 2010), mostly coming with own funding (Cavalletti, Kuwada, Maas, Ohta) attracted by the outstanding reputation of Bonn. ’Optimal transport’ is a crossroads of various areas of mathematics, leading to new deep insights and powerful applications, e.g. in the analysis on Wiener spaces [G:FSS10] or the construction of the heat flow on metric measure spaces [G:OS09] satisfying the curvature-dimension condition of Lott-Sturm-Villani. Kuwada, long-term visitor at HCM with Japanese fellowship, and Philipowski [G:KP11] derived the fundamental monotonicity formulas for Ricci flows through pathwise couplings of Brownian motions w.r.t. Perelman’s $\mathcal{L}$-functional. BIGS-student Weber [G:Web10] succeeded in proving exponential concentration of the invariant measure of the stochastic Allen-Cahn equation in the low temperature regime around the deterministic minimizers. An important breakthrough in the analysis of stochastic processes on the Wasserstein space was the proof of quasi-invariance of the entropic measure in d = 1 by von Renesse and Sturm [G:vRS09], being of independent interest as a Girsanov-type theorem for a-stable processes.

## Dynamics in random environments.

The mathematical analysis of spin glasses, both in its static and dynamics aspects has been one of the major success stories in probability theory over the last decade. On the equilibrium side, the proof of the Parisi conjecture by Talagrand is considered a major achievement. The group around Bovier has successfully focused on the dynamics aspects of mean field spin glasses, often termed ageing. In their work, ageing is obtained by proving the convergence of the clock process associated to the dynamics to stable Lévy subordinators of index $\alpha < 1$. The first result for the case of models with correlated disorder (the p-spin SK models) was obtained by Ben Arous, Bovier, and Cerný [G:BABC08], where convergence of the clock was shown in law; later Bovier and Véronique Gayrard, a frequent visitor to HCM and senior participant in the Junior Trimester Program ‘Stochastics’ at HIM, extended this to almost sure convergence. During this program BIGS student Adela Svejda started to work on the zero-temperature limit of this process. Regular visits, e.g. during the above mentioned Trimester Program by Arguin were crucial for the complete analysis of the extremal process of Branching Brownian Motion, together with Bovier and Kistler (first results are published in [G:ABK]) and two more preprints are available on arXiv). These topics were prominently present in the Trimester Program ‘Complex Stochastic Systems: Discrete vs. Continuous’ and the Junior Trimester Program ‘Stochastics’.

## Metastability.

The potential theoretic approach to metastability developed over the last decade in the group around Bovier has been pushed beyond its original domain of applicability in various directions. Bovier, den Hollander (Bonn Research Chair 2011), and Spitoni obtained sharp estimates on nucleation times in conservative lattice gas models at low temperatures and very large volumes [G:BdHS10]. This made essential use of a variational principle for capacities that allows to produce efficient lower bounds. This has also been used to analyze the distribution of the metastable exit times in the random field Curie-Weiss model by Alessandra Bianchi, Bovier, and Ioffe (a regular visitor at HCM) [G:BBI]. Another important step was done by proving the Eyring-Kramers formula for the stochastic Allen-Cahn equation. This is part of the thesis project of Barret (at École Polytechnique) who visits the HCM regularly to collaborate with Bovier.

## Stochastic algorithms and probabilistic models in economics.

The joint  appointment of Ankirchner (Bonn Junior Fellow) in the mathematics and economy departments has strengthened the link between probability theory and mathematical modeling in economics. The focus within Research Area G is on “continuous” models, based on concepts from stochastic analysis. Objects of investigation have been equilibria in random dynamic games in continuous time, where agents’ decisions interfere with opponents’ payoff profiles. Ankirchner in a joint work [G:AS11] with prospective HCM-postdoc Strack (Research Area I) seized on a contest game in which agents compete in stopping processes at a level as high as possible. They derived optimal stopping times by appealing to nonlinear integral  representation as provided by backward stochastic differential equations, and they obtained time bounds up to which the Nash equilibrium distribution can be embedded. Albeverio has developed new techniques for mathematical models of financial markets with interacting assets [G:ASW07]. Stochastic analysis has been instrumental in addressing economically motivated convergence and stability problems in game theory. Imhof successfully continued his detailed study of stochastic asymptotic stability of diffusion processes describing evolutionary game dynamics [G:HI09]. Stochastic algorithms play a prominent rôle in numerics as well as for theoretical aspects in economics and natural sciences. Karpinski analyzed path coupling and its relationship to stopping times of Markov chain algorithms [G:BDK08]; he also considered Metropolis Glauber dynamics for sampling proper coloring of regular trees. Eberle studied sequential Markov Chain Monte Carlo samplers in continuous time. As a first step towards non-asymptotic bounds under local mixing conditions, he analyzed stability of nonlinear flows of probability measures related to sequential MCMC methods. Jointly with Marinelli [G:EM10] he derived Lp-Lq estimates for time-inhomogeneous transition operators of Feynman-Kac type, applying weighted Poincaré and logarithmic Sobolev inequalities.