# Colloquium in probability and other talks in summer term 2012

**Organisers**: Nina Gantert (TUM), Hans-Otto Georgii (LMU), Franz Merkl (LMU), Silke Rolles (TUM), Vitali Wachtel (LMU), Gerhard Winkler (Helmholtz Zentrum München)

**Talks**:

**Wednesday**, 9th May 2012, 16:30, room 2.01.10, Parkring 11, Garching-Hochbrück

Pierre Mathieu, University Marseille

Title: Renewal theory for random walks on planar hyperbolic groups. Abstract: we use the automatic structure of hyperbolic groups (Series 1981; Cannon, 1984) to construct a regeneration structure for random walks on planar hyperbolic groups. From this regeneration structure, one may recover already known results, such as the law of large numbers or the central limit theorem, but also prove new ones, namely the analyticity of the rate of escape and the asymptotic variance of the walk. Joint work with Peter Haïssinsky and Sebastian Müller (LATP, Université d'Aix-Marseille).

**Tuesday**, 15th May 2012, 16:30, room 2.01.10, Parkring 11, Garching-Hochbrück

Michael Scheutzow, TU Berlin

Title: Moments of recurrence times for Markov chains

Abstract: It is well-known that if for an irreducible discrete time Markov chain with countable state space, the recurrence time for some state has a finite first moment, then the same is true for every state. Kai Lai Chung showed in the 1950-es that the same is true for all polynomial moments and asked for which general moments -- defined via an increasing function $f$ -- this property holds true. We provide an explicit description of all positive non-decreasing functions for which the property holds. This is joint work with Frank Aurzada, Hanna Döring and Marcel Ortgiese.

**Tuesday**, 5th June 2012, 16:30, room 2.01.10, Parkring 11, Garching-Hochbrück

Dr. Patrick Dondl, Technische Universität München

Title: Pinning and depinning of interfaces in random media

Abstract: We consider the evolution of an interface, modeled by a parabolic equation, in a random environment. The randomness is given by a distribution of smooth obstacles of random strength. To provide a barrier for the moving interface, we construct a positive, steady state supersolution. This construction depends on the existence, after rescaling, of a Lipschitz hypersurface separating the domain into a top and a bottom part, consisting of boxes that contain at least one obstacle of sufficient strength. We prove this percolation result. Furthermore, we examine the question of existence of a solution propagating with positive velocity in a random field with non-bounded random obstacle strength. This work shows the emergence of a rate independent hysteresis in systems subject to a viscous microscopic evolution law through the interaction with a random environment.

**Monday**, 11th June 2012, 16:30, room 2.01.10, Parkring 11, Garching-Hochbrück

Ghurumuruhan Ganesan, Indian Statistical Institute, New Delhi, India

Title: Concentration estimates for Stabilizing Functionals of Poisson processes

Abstract: In this talk, we study concentration properties of stabilizing functionals of Poisson processes. Let *Ν* denote a realization of a homogenous Poisson point process in **R**^{d} with intensity λ. For a real valued function f ∈ **L**^{p} that stabilizes at a sufficiently large rate and for a convex compact set *W*, we obtain sharp estimates for the concentration of the sequence

around its mean as n → ∞. As an illustration, we show that parameters in Germ-grain models like Voronoi Tessellation and Radial Spanning Tree stabilize at rate α for every α > 0 and apply our results to obtain the rate of convergence for the corresponding estimators. We finally show how our proof technique also extends to functionals of Binomial processes.

**Tuesday**, 12th June 2012, 16:30, room 2.01.10, Parkring 11, Garching-Hochbrück

Prof. Dr. Achim Klenke, Johannes Gutenberg-Universität Mainz

Title: Spanning trees on ladder-like graphs

Abstract: Consider two copies of the integer lattice and connect any two adjacent points with an edge (a rung) to end up with a doubly infinite ladder. Pick a spanning tree of that graph at random. What is the probability that a given rung is in that tree? Consider the process of rungs in the random spanning tree. This process is a stationary determinantal point process that is also a renewal process. In our second example, we consider three copies of the integer lattice, shift them by 0, 1/3 and 2/3, respectively and connect the vertices in a helix-like fashion (see figure).

(Figure: Spanning tree on the helix graph. For illustration, the circular bonds are colored red and the horizontal bonds are colored blue.) The point process on the circular bonds is a determinantal point process and is a renewal process of order two. In this talk, we present some of the tools, like electrical networks, combinatorics and dynamical systems.

**Wednesday**, 20th June 2012, 17:15, room 2.01.10, Parkring 11, Garching-Hochbrück

Dr. Alexander Schnurr, Technische Universität Dortmund

Title: Fine Properties of Stochastic Processes used in Mathematical Finance

Abstract: We use the symbol and generalizations of the Blumenthal-Getoor index in order to prove growth- and Hölder-conditions as well as maximal inequalities for homogeneous diffusions with jumps. As examples we consider Lévy driven SDEs and the COGARCH process.

**Monday**, 2nd July 2012, 16:30, room B 133, Theresienstr. 39, Munich

Prof Dr. Kosta Panagiotou, Ludwig-Maximilians-Universität München

Title: Catching the *k*-NAESAT Threshold

Abstract: The best current estimates of the thresholds for the existence of solutions in random constraint satisfaction problems ('CSPs') mostly derive from the first and the second moment method. Yet apart from a very few exceptional cases these methods do not quite yield matching upper and lower bounds. According to deep but non-rigorous arguments from statistical mechanics, this discrepancy is due to a change in the geometry of the set of solutions called condensation that occurs shortly before the actual threshold for the existence of solutions. To cope with condensation, physicists have developed a sophisticated but non-rigorous formalism called Survey Propagation, which yields precise conjectures on the threshold values of many random CSPs. In this talk I will discuss a new Survey Propagation inspired method for the random *k*-NAESAT problem, which is one of the standard benchmark problems in the theory of random CSPs. This new technique allows us to overcome the barrier posed by condensation rigorously, and prove very accurate estimates for the *k*-NAESAT threshold; in particular, we verify the statistical mechanics conjecture for this problem. This is joint work with Amin Coja-Oghlan.

**Monday**, 9th July 2012, 16:30, room B 251, Theresienstr. 39, Munich

Franziska Hiermeyer, Ludwig-Maximilians-Universität München

Title: Critical Galton-Watson Processes in a Random Environment

Then subsequently:

Jetlir Duraj, Ludwig-Maximilians-Universität München

Title: Random Walks in Cones, the case of nonzero drift

**Monday**, 16th July 2012, 16:30, room 2.01.10, Parkring 11, Garching-Hochbrück

Dr. Oliver Kley, Friedrich-Schiller-Universität Jena

Title: Entropy of convex hulls and the probability of small balls

Abstract: In the context of empirical processes, R. M. Dudley(1987) raised the question of estimating the covering numbers of the convex hull of a set *T* in Hilbert space relying on information about the covering numbers of the spanning set *T*. This issue was recognized to be of relevance in diverse settings in functional analysis, approximation theory, geometry as well as in probability theory. The talk tends to emphasize the tight connection of the problem to Gaussian stochastic processes and quantities related to them such as reproducing kernel Hilbert spaces. In particular, we will give a probabilistic and unified proof for the estimates of covering numbers of convex hulls in the cases of interest using small ball probabilities. Vice versa, we obtain small ball probabilities from convex hull estimates.

How to get to Garching-Hochbrück