# Oberseminar Wahrscheinlichkeitstheorie und andere Vorträge im Sommersemester 2014

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

**Talks**:

**Wednesday**, 5^{th} March 2014,16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück ( Technische Universität München)

Dr. Alexandre Gaudilliere (Universität Marseille)

Title: On some random forests with determinantal roots.

**Monday**, 10^{th} March 2014,16:00, LMU, room B 251, Theresienstr. 39, Munich

Shirin Riazy (LMU, Mathematisches Institut der Universität München)

Title: Percolation processes on models of neural networks.

**Monday**, 10^{th} March 2014,17:00, LMU, room B 251, Theresienstr. 39, Munich

Matthias Hofer (LMU, Mathematisches Institut der Universität München)

Title: Summenzerlegungen und integrierte Irrfahrten.

**Monday**, 7^{th} April 2014,16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Francoise Pene (Universität Brest, France)

Title: Limit theorems for U-statistics indexed by a random walk.

Abstract: We establish results of convergence in distribution for (suitably normalized) U-statistics indexed by a random walk. We will explain the different behaviours and compare them with those of random walks in random scenery. This work is in collaboration with Brice Franke and Martin Wendler.

**Monday**, 28^{th} April 2014,16:30, LMU, room B 251, Theresienstr. 39, Munich

Dr. Iulia Dahmer (Universität Frankfurt)

Title: Keywords: Kingman coalescent, internal branch lengths, site frequency spectrum, coupling.

Abstract: The Kingman coalescent is a classical population genetics model for describing the genealogy of (haploid) populations in equilibrium. If the population size is n, the corresponding coalescent can be graphically represented as a binary rooted tree started with n leaves. The length Ln,r of order r in the Kingman n-coalescent tree is defined as the sum of the lengths of all branches that support r leaves. For r = 1 these branches are external, while for r 2 they are internal and carry a subtree with r leaves. We prove that for any s 2 N the vector of centred and rescaled lengths of orders 1 r s converges to the multivariate standard normal distribution as the number of leaves of the Kingman coalescent tends to infinity. To this end we use a coupling argument for Markov chains which shows that for any r 2 the (internal) length of order r behaves asymptotically in the same way as the length of order 1 (the external length). A direct consequence of our result is that the first finitely many components of the site frequency spectrum associated with the population are asymptotically independent Poisson-distributed random variables. On the other hand, our result plays a central role in the description of the dynamics of the process of external length of an evolving Kingman coalescent and we will give also insight on this matter. This talk builds on joint work with Götz Kersting.

**Monday**, 5^{th} May 2014

Graduate Seminar Financial- and Actuarial Mathematics LMU and TUM

**Monday**, 12^{th} May 2014,16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Prof. Jonathon Peterson (Purdue Universität)

Title: Extreme slowdowns for excited random walks.

Abstract:Excited random walks (also called cookie random walks) are a model for self-interacting motion where the transition probabilities are governed by the local time of the random walk at the current site. If the excited random walk has a positive limiting speed $v_0 = \lim_{n\rightarrow \infty} X_n/n$, then for $v \in (0,v_0)$ the events $\{X_n < n v\}$ and $\{T_n > n/v\}$ are "slowdown events." In a previous study of the large deviations for excited random walks, it was shown that the probabilities of these slowdown events decays at a polynomial rate that can be explicitly calculated. In this talk we consider the asymptotics of the probabilities of the extreme slowdown events $\{X_n < n^\gamma\}$ and $\{T_{n^\gamma} > n \}$ with $\gamma < 1$. We compute precise estimates on the polynomial rate of decay for these events and show that an interesting transition occurs at $\gamma = 1/2$.

**Monday**, 19^{th} May 2014,16:30, LMU, room B 251, Theresienstr. 39, Munich

Markus Kaiser (LMU)

Title: Asymptotics of one-dimensional Brownian motion among Poissonian obstacles**.**

Abstract:We use Feynman-Kac formula to analyze the asymptotical probability of surviving of one-dimensional Brownian motion in a random environment dosed with hard obstacles following an Poissonian law, i.e. we have a look at the probability that a Brownian motion stays up to some time s within an random interval, where the boundaries of the interval are given by the values of two exponentially distributed random variables, and then send s to infinity. We further investigate the law one obtains by conditioning on the event that Brownian motion does not leave the interval up to a fixed point in time s by using Methods from stochastical analysis, in particular the Girsanov Theorem.

**Monday**, 26^{th} May 2014

Graduate Seminar Financial- and Actuarial Mathematics LMU and TUM

**Monday**, 23^{th} June 2014, 16:30 and 17:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück.

16:30, Stefan Junk (Technische Universität München)

Title: Survival Probability for a random walk in a disastrous random environment.

Abstract: Directed Polymers in random environments (DPREs) are a model for paths in Z^d. The environment consists of i.i.d. real random variables at each space-time point, where the path is attracted to positive values and repelled by negative values, and the probability for each path is proportional to the values picked up along its run. There is a natural martingale for this quantity having an almost sure limit, which is either zero or positive depending on the strength of the environment, and an interesting transition occurs between the two phases. We will first give an overview over results and open problems for this model.

Furthermore we present a model for a random walk in a disastrous environment. Here the environment consists of independent Poisson processes at each site of Z^d, the jump times of which are interpreted as a disaster for this site. A continuous time simple random walk is killed once such a disaster occurs at the site it currently occupies. The probability of surviving up to time t as a random variable of the environment can be seen as continuous time analog of DPREs in a degenerate environment. A number of results from DPREs have a correspondence for this model, most importantly that the martingale limit is almost surely zero in dimension d=1 and 2.

17:30, Prof. Christopher Hoffman (University of Washington, Washington, USA)

Title: Pattern Avoiding Permutations and Brownian Excursion

Abstract: Permutations that avoid a given pattern are some of the most classical objects in combinatorics. We will consider the shape of a randomly chosen 231-avoiding or 321-avoiding permutation. We will show that these have a strong connection to Brownian excursion.

This is joint work with Douglas Rizzolo and Erik Slivken

**Monday**, 30^{th} June 2014,16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Prof. Dr. Patrick Dondl (Universität Durham)

Title: Ballistic propagation of interfaces in random media with unbounded obstacles.

Abstract: We consider the depinning phase transition in the propagation of discrete interfaces in random environments. The evolution equation, a variant of the Quenched Edwards-Wilkinson model, is a semilinear parabolic lattice differential equation with a constant driving term and a random nonlinearity to model the influence of the environment. For the case where this random field consists of obstacles admitting a random strength with bounded exponential moment, we show that the interface propagates with a finite velocity for sufficiently large driving force. The proof uses a supermartingale estimate akin to the study of branching random walks.

**Monday**, 30^{th} June 2014

Graduate Seminar Financial- and Actuarial Mathematics LMU and TUM

How to get to Garching-Hochbrück