# Colloquium in probability and other talks in winter term 2011/12

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

**Talks:**

- Monday, 17. October 2011, Garching-Hochbrück, 16:30 pm, room 2.01.10.

Prof. Dr. Makiko Sasada, Keio University

Abstract: We study a class of exclusion processes which can be understood as an intermediate between SSEP and TASEP. For these processes, it is proved that the particle density converges to the solution of the nonlinear diffusion equation under the diffusive rescaling in space and time. The relation between macroscopic behavior of these processes and that of SSEP or TASEP will be discussed.

- Monday, 31. October 2011, Garching-Hochbrück, 16:30 pm, room 2.01.10.

Prof. Dr. Wolfgang Woess, Technische Universität Graz, Austria

Abstract: Consider a proper metric space X and a sequence (F_n) of i.i.d. random continuous mappings of X onto itself. It induces the stochastic dynamical system (SDS) X_n^x = F_n(X_{n-1}^x)) starting at X_0 = x in X. We study existence and uniqueness of invariant measures, as well as recurrence and ergodicity of this process. In the first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying stochastic iterations. In the second part, we consider the case where the F_n are Lipschitz mappings. The main results concern the critical case when the associated Lipschitz constants are log-centered. Prinicpal tools are the Chacon-Ornstein theorem and a hyperbolic extension of the space X as well as of the process (X_n^x). An example where the results apply is the reflected affine stochastic recursion given by X_n^x = |A_nX_{n-1}^x - B_n|, where (A_n,B_n) is a sequence of two-dimensional i.i.d. real random variables with A_n > 0.

- Monday, 7. November 2011, Garching-Hochbrück, 16:30 pm, room 2.01.10.

Prof. Dr. Piotr Sniady, Universität Wroclaw, Polen und Technische Universität München

Title: Second class particles and random Young tableaux (joint work with Dan Romik)

Abstract: We study a totally antysymmetric exclusion process with transition probabilities related to the representation theory of the symmetric groups (Plancherel measure). We show that in this model the speed of the second class particle coverges almost surely to a limit which is a random variable with a known distribution (analogous result for TASEP is due to Mountford and Guiol). Since our model is closely related to the representation theory, we can take advantage of some additional structures; we can also apply our result in the harmonic analysis on the infinite symmetric group and answer some old questions from this theory

- Monday, 14. November 2011, Mathematical Institute of LMU München, 16:15 pm, room B 251.

Dr. Mathias Rafler, Technische Universität München

In this context we would like to refer to our workshop on pointprocesses and indistinguishable particles:

http://www-m5.ma.tum.de/Allgemeines/PPundSUT

Title: A Bayesian view on the Polya sum process

Abstract: The Polya sum process was introduced recently by H. Zessin as the process which replaces the iid mechanism of the Poisson process by a Polya urn mechanism. A different class of point processes are the Cox processes, Poisson processes with a random intensity measure, whose distribution in Bayesian terms is the priori measure. We study a particular family of Gamma processes and show that this is a conjugate family of directing measures. Moreover, the Bayes estimator turns out to be the Polya sum kernel.

- Monday, 21. November 2011, Garching-Hochbrück, 16:30 pm, room 2.01.10.

Dr. Bálint Vető, Universität Bonn

Title: Diffusive behaviour of the myopic self-avoiding walk in three or more dimensions (joint work with Illes Horvath and Balint Toth)

Abstract: The myopic (or `true') self-avoiding walk was introduced in the physics literature in the 1980's as a natural model of self-repellence. The random walker on**Z**^{d}is pushed to areas that were less visited in the past. More precisely, the jump rates depend on the negative gradient of the local time of the walker. We investigate the asymptotic behaviour in three and more dimensions. For a wide class of self-interaction functions, we identify a natural time-stationary distribution of the local time profile as seen from the moving particle. We prove diffusive upper and lower bounds on the displacement. For a more specific class of interactions, we establish central limit theorem for the finite dimensional distributions of the displacement. In the proof of the central limit theorem, we use the non-reversible version of the Kipnis-Varadhan theory.

- Monday, 28. November 2011, Garching-Hochbrück, 16:30 pm, room 2.01.10.

Dr. Artem Sapozhnikov, ETH Zürich

Title: Connectivity properties of random interlacements

Abstract: In this talk, we consider the interlacement Poisson point process on the space of doubly-infinite**Z**^{d}-valued trajectories modulo time-shift, tending to infinity at positive and negative infinite times, introduced by Sznitman. The set of edges traversed by at least one of these trajectories induces the random interlacement graph (at level u), an infinite connected subgraph of**Z**^{d}. We will summarize recent results about connectivity properties of the random interlacement graph including the transience of the graph and the non-trivial Bernoulli phase transition. Our results are valid for dimensions d>=3 and levels u>0. The talk is based on joint work with Balazs Rath (ETH, Zurich).

- Monday, 5. December 2011, Garching-Hochbrück, 16:30 pm, room 2.01.10.

Dr. Martin Hutzenthaler, LMU München

Title: Branching diffusions in random environment

Abstract: Individuals in a branching process in random environment (BPRE) branch independently of each other and the offspring distribution changes randomly over time. The variation in the offspring distribution models fluctuations in environmental conditions. Here we study the diffusion approximation of BPREs, which we denote as branching diffusion in random environment (BDRE). The one-dimensional BDRE (modelling a panmictic population) turns out to have strong analytical properties similar to Feller's branching diffusion. We obtain a phase transition for the survival probability in the subcritical regime which is well-known for BPREs. In addition we establish a phase transition in the super-critical regime which has not been reported for BPREs yet. Finally we analyse the long-time behavior of interacting branching diffusions in random environment (modelling structured populations).

- Monday, 12. December 2011, Mathematical Institute of LMU München, 16:15 pm, room B251.

Sebastian Carstens, LMU München

Title: Eindeutigkeit und Nicht-Koexistenz von unendlichen Clustern in zweidimensionaler abhängiger Perkolation.(arXiv:1103.0901v2)

Abstract: Nach einer kurzen Einführung in die Perkolation wiederholen wir den bekannten Satz von Burton und Keane über die Eindeutigkeit von unendlichen Clustern und skizzieren den eleganten Beweis. Danach stellen wir kurz „Zhang's argument“ vor, welches bei Rotationsinvarianz die Koexistenz von unendlichen Clustern verhindert. Als letztes zeigen wir, warum dieses Resultat auch ohne Rotationsinvarianz gültig bleibt.

- Wednesday, 21. Dezember 2011, Garching-Hochbrück, 16:30 p.m., Room 2.01.10.

Prof. Dr. Wolfgang König, WIAS und TU Berlin

Title: Eigenvalue order statistics and mass concentration in the parabolic Anderson model

Abstract: We consider the random Schrödinger operator on the lattice with i.i.d.~potential, which is double-exponentially distributed. In a large box, we look at the lowest eigenvalues, together with the location of the centering of the corresponding eigenfunction, and derive a limit law, after suitable rescaling and shifting, towards an explicit Poisson point process. This is a strong form of Anderson localisation at the bottom of the spectrum. Since the potential is unbounded, also the eigenvalues are, and it turns out that the gaps between them are much larger than of inverse volume order. We explain an application to concentration properties of the corresponding Cauchy problem, the parabolic Anderson model. In fact, it will turn out that the total mass of the solution comes from just one island, asymptotically for large times. This is joint work in progress with Marek Biskup (Los Angeles and Budweis)

- Monday, 9. January 2012, Mathematisches Institut der LMU München, 16:15 Uhr, Raum B 251

Prof. Dr. Götz Kersting, Universität Frankfurt

Title: Coalescent trees and their symmetries

Abstract:**N**-Coalescent trees describe the genealogy of**N**individuals chosen at random from a large population. Important quantities (from the point of view of applications) are their total lenght*L*_{N}as well as their total external length*E*_{N}, that is the total length of all branches ending in a leaf. In the talk we are concerned with the asymptotic distribution of these quantities. We shall address Kingman coalescents (which models the genealogy of a haploit population in equilibrium with random mating) as well as the larger class of Beta-coalescents. The results use hidden symmetries within these coalescents. Part of the talk builds on joint work with Svante Janson (Uppsala).

- Wednesday, 11. January 2012, Garching-Hochbrück, 16:30 p.m., room 2.01.10

Dr. Markus Heydenreich, Universität Leiden, Niederlande

Titel: A new lace expansion for high-dimensional incipient infinite cluster

Abstract: Incipient infinite cluster (IIC) is a critical percolation cluster conditioned on infinite size. It is a natural example of a spatial object showing fractal properties, self-similarity, and a non-degenerate scaling limit. Kesten (1986) proved its existence on the two-dimensional lattice. This talk concentrates on incipient infinite clusters in high-dimension, whose existence was proven by Van der Hofstad and Jarai (2004). I review the lace expansion for percolation by Hara and Slade (1990), and explain a new lace expansion approach for the backbone of the incipient infinite cluster. This new approach allows the identification of the scaling limit of the IIC backbone. The talk is based on joint work with R. van der Hofstad, T. Hulshof and G. Miermont.

- Monday, 30. January 2012, Garching-Hochbrück, 16:30 p.m., room 2.01.10.

Dr. Chiranjib Mukherjee, Friedrich-Alexander-Universität Erlangen-Nürnberg

Title: Large deviations for Brownian intersection measures

Abstract: We consider a number of independent Brownian motions in a bounded domain in**R**^{d}running until time*t*and look at their intersection set, points which are hit by all the motions before time*t*. There is a measure*l*_{t}on the intersection set which counts the intensity of the path intersections and is called the Brownian intersection measure. We derive large-*t*asymptotics of this measure in terms of a large deviation principle in the set of finite measures on B. The rate function is explicit and gives some direct meaning of the intersection measure as the pointwise product of the individual occupation measures. This is, in fact, an extension of Donsker-Varadhan large deviations to a non-linear setting and it shows that, the intersection measure is asymptotically of finite order and admits a nice shape. A second version of our principle is proved for the motions observed until the individual exit times from the domain, conditional on a large total mass in some compact sub-domain. This extends earlier studies on the intersection measure by König and Mörters. This is joint work with Wolfgang König (Berlin).

- Monday, 6, February 2012, Garching-Hochbrück, 16:30 p.m., room 2.01.10

Prof. Dr. Peter Pfaffelhuber, Universität Freiburg

Title: The evolution of genealogical trees under selection

Abstract: We construct a tree-valued Markov process describing the evolution of genealogical relationships in populations of constant size. While the case of equally fit individuals is termed neutral evolution, we concentrate on the case where the distribution in offspring numbers depend on the type of an individual. In the talk we will develop similarities to the measure-valued Fleming-Viot process with mutation and selection, which will also be explained in detail. This is joint work with Andrej Depperschmidt and Andreas Greven.

- Monday, 20, February 2012, Garching-Hochbrück, 16:30 p.m., room 2.01.10

Dr. Matthias Meiners, Universität MünsterHow to get to Garching-Hochbrück