# Colloquium in probability and other talks in summer term 2013

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

**Talks**:

**Monday**, 15th April 2013, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Dr. Chiranjib Mukherjee (Technische Universität München, Germany)

Title: Convergence of path measures with mean-field type interactions

Abstract: We consider long time behavior of Gibbs measure on three dimensional Brownian paths with mean-field type Hamiltonian with Coulomb interaction. The free energy admits a well-studied variational formula (Donsker-Varadhan [1983], Lieb [1976]). It turns out that the long time asymptotics of the path measures converges to a mixture of the minimizers of the free energy variational formula. This model is related to the Polaron problem, where the behavior of the path measures (in a certain regime known as 'strong coupling') is of interest and possesses a number of open problems.

This is joint work (in progress) with Erwin Bolthausen (Zürich).

**Monday**, 6th May 2013

Oberseminar Finanz- und Versicherungsmathematik

**Monday**, 13th May 2013, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Dr. Christian Döbler (Technische Universität München, Germany)

Title: Stein's method and simple random walk statistics

Abstract: We show how Stein's method can be used to derive rates of convergence in several quite counter intuitive limit theorems related to simple random walk on the integers. Precisely, we derive concrete error bounds on the approximation of the relative time spent positive by simple random walk by the arcsine distribution and on the approximation of the maximum value, the number of returns to the origin and the number of sign changes up to a given time by the half-normal distribution. This is done by comparing the characterization of the limiting distribution with that of the approximating discrete distribution, exploiting a recent technique by Goldstein and Reinert. Finally, we hint at some possible future applications of this method.

**Monday**, 3rd June 2013

Oberseminar Finanz- und Versicherungsmathematik

**Monday**, 10th June 2013, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Dr. Lorenz Gilch (Technische Universität Graz)

Title: Asymptotic Entropy of Random Walks on Regular Languages

Abstract: In this talk I will present my recent results about asymptotic entropy and its properties of random walks on regular languages over a finite alphabet. In particular, this setting applies to the case of random walks on virtually free groups. Existence of the asymptotic entropy is shown and formulas for it are presented. Moreover, I will show that the entropy is the rate of escape with respect to the Greenian metric and varies analytically in terms of probability measures of constant support.

**Monday**, 17th June 2013, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Dr. Omar Boukhadra (Université de Constantine)

Title: Random conductances model with polynomial tail and the four-dimensional case

Abstract: The talk first concerns the decay of the return probabilities of random walks among polynomial lower tail i.i.d. random conductances on the d-dimensional integer lattice. We show that when the dimension is larger than 4 and the power of the lower tail is larger than 1/4, we have a standard behavior. This is a work in progress with Pierre Mathieu

(Marseille). In a previous work, we showed that when the power is small enough, the behavior is anomalous. These mean that the transition from regular decay to anomalous decay in dimensions larger than 5 actually occurs in the class of power-law tails. Second, it is about the four-dimensional case. Marek Biskup (UCLA) and I showed that it is possible to construct a random environment governed by i.i.d. random conductances for which there is an anomalous behavior that results in a logarithmic term, which supports an upper bound proved by Berger, Biskup, Hoffman and Kozma (2008).

**Monday**, 1st July 2013, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Prof. Dr. Dmitry Ioffe (Technion - Israel Institute of Technology)

Title: An invariance principle for random walks with prewetting

Abstract: The abstract: I shall discuss an invariance principle for a class of 1+1 SOS interfaces (walks) in potential fields. The interfaces are constrained to stay above the wall. The potential field is λ X^{a} with a ≥ 1. The case a=1 corresponds to tilting areas below interfaces. Limiting objects, as λ tends to zero, are reversible diffusions with drifts given by log-derivatives of ground states for the associated singular Sturm-Liouville operators. For a=1 such ground state is a rescaled Airy function, and the corresponding diffusion was derived by Ferrari and Spohn in the context of scaling limits for Brownian motions conditioned to stay above circular barriers.

Based on a work in progress with Senya Shlosman and Yvan Velenik.

**Friday**, 5th July 2013, and **Saturday**, 6th July 2013

Mini-Workshop "Women in Probability"

**Monday**, 8th July 2013, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Dr. Konrad Kolesko (Uniwersytet Wrocawski, Poland)

Title: Linear stochastic equations in the critical case

Abstract: In this talk we will focus on the fixed point equation of inhomogeneous smoothing transformation. We are particular interested in the problem of existing and tail behaviour of a solution in the critical case. We will also show some relations with the rightmost point ever reached by a particle of the underlying branching process.

**Monday**, 8th July 2013

Oberseminar Finanz- und Versicherungsmathematik

**Monday**, 15th July 2013, 16:00 and 17:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Prof. Dr. Tomohiro Sasamoto (Technische Universität München, Germany)

Title: The KPZ scaling functions for a two-component exclusion process

Abstract: The scaling functions which appear in the studies of models in the KPZ universality class are considered to be universal. The most typical example is the Tracy-Widom distributions which describe the fluctuations of the height in growth models or the current in driven lattice gases. Another example is the stationary two point function first studied by Pr"ahofer and Spohn. In a recent paper, H. Beijeren argued that the same universal scaling function would appear in more generic one-dimensional fluids. His arguments are based on mode-coupling approximation and have not been verified. In this presentation,we consider a two-component exclusion process and discuss a possibility of observing the same scaling function. After short review of the one-component case, we explain the non-linear fluctuating hydrodynamical description of the process (coupled KPZ equation), the predictions based on the scheme and the knowledge on the exact stationary measure, and the comparison with the Monte-Carlo simulations. The main purpose of the presentation is to explain the new developments in the area, provide some evidence (though numerical for now) and propose a possible direction for more rigorous investigations. This is based on a collaboration with P.L. Ferrari and H. Spohn.

Prof. Dr. Patrik Ferrari (Institut für Angewandte Mathematik, Universität Bonn)

Title: Anomalous shock fluctuations in TASEP and last passage percolation models

Abstract: We consider the totally asymmetric simple exclusion process with initial conditions and/or jump rates such that shocks are generated. If the initial condition is deterministic, then the shock at time *t* will have a width of order *t*^{1/3}. We determine the law of particle positions in the large time limit around the shock in a few models. In particular, we cover the case where at both sides of the shock the process of the particle positions is asymptotically described by the Airy_{1} process. The limiting distribution is a product of two distribution functions, which is a consequence of the fact that at the shock two characteristics merge and of the slow decorrelation along the characteristics. We show that the result generalizes to generic last passage percolation models.

This is joint work with Peter Nejjar.

**Monday**, 2nd September 2013, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Akram Chawki (Ludwig-Maximilians-Universität Munich)

Title: Zufällige Verflechtungen

Abstract: In diesem Vortrag wird das Modell der zufälligen Verflechtungen ("Random Interlacements") vorgestellt, in dem durch einen Poisson Punktprozess zufällig ausgewählte, beidseitig unendliche Pfade auf **Z*** ^{d}* betrachtet werden. Durch einen nicht negativen Parameter

*u*wird angegeben, wie viele Pfade gleichzeitig vorkommen. In Abhängigkeit von

*u*wird dann das Perkolationsverhalten des Komplements dieser Pfade untersucht. Das kann dann benutzt werden um die Zeit abzuschätzen, zu der der diskreten Zylinder (

**Z**/

*N*

**Z**)

^{d}×

**Z**für große

*N*durch die einfache Irrfahrt in zwei unendliche Teile getrennt wird.

**Thusday**, 10th September 2013, 15:00 and 16:30, LMU, Theresienstr. 39, Munich

15:00, room A 348, Katja Miller (TUM and Ludwig-Maximilians-Universität Munich)

Title: Directed Random Walks on Percolation Clusters with Carrying Capacities

Abstract: We consider a directed random walk on an oriented percolation cluster. If the walk is restricted to the backbone of the infinite cluster, Birkner et al., 2012, showed that we get a law of large numbers and an annealed and quenched central limit theorem. In this thesis we equip all sites in the open cluster with a strictly positive random number and have the random walk choose its next step with probability proportional to these carrying capacities. It is possible to analyse this weighted random walk with the same methods as in the original paper, if we do not change the shape of the percolation cluster and choose the carrying capacities appropriately. We show that a law of large numbers and an annealed central limit theorem hold if the carrying capacities are strongly mixing in the time coordinate. Furthermore, a quenched central limit theorem holds if the carrying capacities are finitely dependent in space and time coordinates. Finally we discuss the results for some examples.

16:30, room B 349: Alexisz Gaál (Ludwig-Maximilians-Universität Munich)

Title: Spontaneous breaking of rotational symmetry in a probabilistic hard disk model in statistical mechanics

Abstract: In this talk I will present a simple model of a two-dimensional crystal with hard core interaction in which the rotational symmetry is spontaneously broken in a strong sense. This model is restricted to a simple phase space of planar point configurations. Every admissible configuration in this phase space is a perturbed version of some standard triangular lattice with side length one. I will show that one can find an infinite volume Gibbs measure such that the expected distance from a fixed standard triangular lattice is small.

** Monday,** 23rd September 2013, 16:30, LMU, room B 349, Theresienstr. 39, Munich

Michael Bär (Ludwig-Maximilians-Universität Munich)

Title: Kolmogorov Diffusion conditioned to be positive

Abstract: The Kolmogorov diffusion is a two dimensional real valued stochastic process with an integrated Brownian motion as first coordinate and its integrand Brownian motion as second coordinate. In this talk I will present some general facts about this process and then focus on the Kolmogorov diffusion with the first coordinate conditioned to be positive started at the origin (0,0). I will show that this conditioned process exists as a limit by demonstrating convergence of transition densities and tightness of the according family of probability measures. In order to do so I will calculate an explicit form of the transition density.

How to get to Garching-Hochbrück