# Oberseminar Wahrscheinlichkeitstheorie und andere Vorträge im Wintersemester 2012/13

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

**Talks**:

**Friday**, 19th October 2012, 15:00, LMU, room B 251, Theresienstr. 39, Munich

Dr. Jean-Christophe Mourrat (Ecole Polytechnique Federale de Lausanne, Switzerland)

Title: On the homogenization of the heat equation with random coefficients

Abstract: We consider the heat equation with random coefficients on **Z**^{d}. The randomness of the coefficients models the inhomogeneous nature of the medium where heat propagates. We assume that the distribution of these coefficients is invariant under spatial translations, and has a finite range of dependence. If a solution to this equation is rescaled diffusively, then it converges to the solution of a heat equation with constant coefficients. In probabilistic terms, this convergence corresponds to the fact that the associated random walk satisfies a central limit theorem. I will present recent progress on the estimation of the speed of this convergence, based on the random walk representation.

**Monday**, 22th October 2012, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Dr. Kilian Raschel (Tours University, France)

Title: Random walks in the quarter plane, harmonic functions and conformal mappings

Abstract: In this talk I will present a new approach for finding positive harmonic functions for random walks in a quarter plane. When the drift of the random walk is zero, we prove that there exists a unique harmonic function. In addition, in the non-zero drift case, we rediscover many recent results on harmonic functions (in particular, their number and their explicit expression). The two key-ingredients of this approach are a functional equation satisfied by any harmonic function, and, surprisingly, a certain conformal mapping (introduced in the 90's by Fayolle, Iasnogorodski and Malyshev). Finally we shall see some extensions of our methods.

**Monday**, 29th October 2012, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Dr. Vlad Vysotsky (St. Petersburg State University, Russia)

Title: Persistence of integrated random walks and other random sequences

Abstract: Consider a sequence of oscillating random variables. What is its persistence probability, that is the chance that it stays positive for a long time? We obtain the sharp asymptotic for integrated random walks of a certain type. The problem is deeply related with certain models of mathematical physics. In general, persistance probabilities are well understood only for very few types of random sequences. We give an overview of the field and in addition, discuss our results on persistence of iterated random walks.

**Wednesday**, 7th November 2012, 16:30, TUM, room 2.02.11, Parkring 11, Garching-Hochbrück

Ron Rosenthal (Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel)

Title: Asymptotic behavior of isoperimetric problems in two-dimensional super-critical bond percolation

Abstract: Isoperimetry is a well-studied subject that has found many applications in geometric measure theory, e.g., concentration of measures, heat-kernel estimates, mixing time, etc. Consider the super-critical bond percolation on **Z**^{d} (the d-dimensional square lattice), and Φ_{n} the Cheeger constant of the unique giant component in [-n,n]^{d}. Following several papers which showed that the leading order asymptotics of Φ_{n} is ^{1}/_{n}, Benjamini conjectured that the limit lim_{n→∞} ^{n}/Φ_{n} exists. Restricting to the two-dimensional case, we prove that Φ_{n} scales a.s. to a deterministic quantity, given by the solution to a specific isoperimetric problem on **R**^{2}. The last, is given by a norm defined via one sided boundaries of paths in the percolation graph. In fact we managed to prove much more showing that sets which realize the Cheeger constant converge to the Wulff shape generated by the same norm.

(Joint work with Marek Biskup, Oren Louidor and Eviatar Procaccia)

**Monday**, 12th November 2012, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Tal Orenshtein (Technische Universität München, Germany)

Title: One dimensional Excited Random Walk with a never-ending supply of cookies

Abstract: One dimensional Excited Random Walk has the following amusing description. On each site in the one dimensional lattice there is a pile of enumerated cookies. When the cookie monster arrives a certain site it eats the next available cookie there (and therefore that cookie is no longer there). The excitement it gets by the consumption results in jumping to a neighbor with a drift given by the cookie. The standard assumptions are either that the number of cookies per site is finite (that is, there is no drift after a fixed number of visits) or that the direction of the drift of all cookies is eventually the same. We consider the walk without these assumptions and supply a machinery to determine its asymptotic behavior. As an application, we get an explicit formula for an invariant which determines whether the walk is recurrent, right transient or left transient, for a large family of cookie environments. The proof involves the analysis of a related countable-state Markov chain, which looks locally like a Bessel process. (Joint work with Gady Kozma and Igor Shinkar)

**Monday**, 19th November 2012, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Xiaoqin Guo (Technische Universität München, Germany)

Title: Einstein relation for random walks in balanced random environment.

Abstract: The Einstein relation for a system in equilibrium describes a relation between the response of the system to a perturbation and its diffusivity at equilibrium. It states that the derivative of the velocity (with respect to the strength of the perturbation) equals the diffusivity. In this talk, I will explain the Einstein relation in the context of random walks in a balanced uniformly elliptic iid environment. Moreover, we will provide a new interpretation and a more general version of the Einstein relation in RWRE.

**Monday**, 26th November 2012, 15:00 and 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Prof. Dr. Fabrice Gamboa (Institut de Mathématiques de Toulouse, France), 15:00

Title: Statistical identification for Gaussian processes indexed by a graph

Abstract: In this talk, we will define some Gaussian processes indexed

by a given graph. We will provide and study a Witthle approximation

of the likelihood. This work is motivated by speed prediction on a road

network. (Joint work with Thibault Espinasse and Jean-Michel Loubes)

and

Prof. Dr. Alain Rouault (Université de Versailles-Saint-Quentin, France), 16:30

Title: Truncation of unitary matrices and Brownian bridges

Abstract is available here.

**Monday**, 10th December 2012, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Prof. Dr. Christian Borgs (Microsoft Corporation)

Title: Convergence of sparse graphs as a problem at the intersection of graph theory, statistical physics and probability

Abstract: Many real-world graphs are large and growing. This naturally raises the question of a suitable concept of graph convergence. For graphs with average degree proportional to the number of vertices (dense graphs), this question is by now quite well-studied. But real world graphs tend to be sparse, in the sense that the average or even the maximal degree is bounded by some reasonably small constant. In this talk, I study several notions of convergence for graphs of bounded degree and show that, in contrast to dense graphs, where various a priori different notions of convergence are equivalent, the corresponding notions are not equivalent for sparse graphs. I then describe a notion of convergence formulated in terms of a large deviation principle which implies all previous notions of convergence. (No prior knowledge of graph convergence or large deviation theory is needed.)

**Monday**, 17th December 2012, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Stein Andreas Bethuelsen (Mathematisch Instituut, Universiteit Leiden, The Netherlands)

Title: Dobrushins uniqueness condition for Gibbs measures

Abstract: In the late 60's Ronald Dobrushin presented a simple, but very general condition for showing uniqueness of the Gibbs measures. For many models from statistical mechanics the condition consist of an easy calculation, and shows in this setup the absence of a phase transition for such systems in the high temperature regime. The condition has since it was introduced seen several improvements, most notably the Dobrushin-Shlosman uniqueness condition. Originally the condition was given via coupling methods. I will in this talk however discuss the general idea behind Dobrushins uniqueness condition viewed mainly from a more analytic point of view, enlightening a certain dusting perspective. By again rather simple methods I will show how the Dobrushin and the Dobrushin-Shlosman condition can be improved and put them in a more general framework. Moreover, possible implication of this extension to similar conditions for Interacting Particle Systems will be discussed. (Joint work with Roberto Fernandez and Siamak Taati)

**Monday**, 7th January 2013, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Prof. Mikhail V. Menshikov (University of Durham, United Kingdom)

Title: Explosion, implosion, and moments of passage times for continuous-time Markov Chains: semimartingale approach

Abstract: We establish general theorems quantifying the notion of recurrence - through an estimation of the moments of passage times - for irreducible continuous-time Markov chains on countably infinite state spaces. These theorems are applied to models whose jump processes have a deeply critical behaviour. Sharp conditions of occurrence of the phenomenon of explosion are also obtained. A new phenomenon of implosion is introduced and sharp conditions for its occurrence are proven.

**Monday**, 14th January 2013, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Tim Hulshof (Technische Universiteit Eindhoven, The Netherlands)

Title: Scaling limit of the high-dimensional IIC backbone

Abstract: The incipient infinite cluster (IIC) of percolation is the random subgraph (of **Z**^{d}) that is obtained by conditioning on the event that the origin is part of an infinite cluster at criticality. The IIC contains another random subgraph that is called the backbone. The backbone contains all the vertices in the IIC that have disjoint paths to the origin and to infinity. In high dimensions (typically, when d>6), the backbone resembles a random, singly infinite path. I will discuss recent work in which we prove that the scaling limit of the backbone is a d-dimensional Brownian motion. The proof of this fact relies on a new lace-expansion that closely resembles the lace-expansion for self-avoiding walks. This lace-expansion is fairly simple. (Joint work with Markus Heydenreich, Remco van der Hofstad, and Gregory Miermont)

**Monday**, 21st January 2013, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Prof. Dr. Serguei Popov (University of Campinas - UNICAMP, Brazil)

Title: Soft local times and decoupling of random interlacements

Abstract: We establish a decoupling feature of the random interlacement process **I**^{u} in **Z**^{d} , at level u, for d ≥ 3. Roughly speaking, we show that observations of **I**^{u} restricted to two disjoint subsets A_{1} and A_{2} of **Z**^{d} are approximately independent, once we add a sprinkling to the process **I**^{u} by slightly increasing the parameter u. Our results differ from previous ones in that we allow the mutual distance between the sets A_{1} and A_{2} to be much smaller than their diameters. We then provide an important application of this decoupling for which such flexibility is crucial. More precisely, we prove that, above a certain critical threshold u**, the probability of having long paths that avoid **I**^{u} is exponentially small, with logarithmic corrections for d = 3. To obtain the above decoupling, we first develop a general method for comparing the trace left by two Markov chains on the same state space. This method is based in what we call the soft local time of a chain. In another crucial step towards our main result, we also prove that any discrete set can be "smoothened" into a slightly enlarged discrete set, for which its equilibrium measure behaves in a regular way. (Joint work with Augusto Teixeira)

**Monday**, 28th January 2013, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Ron Doney (School of Mathematics - The University of Manchester, United Kingdom)

Title: Local limit results for subordinators

Abstract: The asymptotic behaviour of first passage times for subordinators is quite different from that for oscillating processes, typically exhibiting exponential decay rather than power-law decay. In this work we improve on the results of Jain and Pruitt (Ann. Probab., 1987) by finding uniform asymptotic estimates for the position of the process under weak assumptions, and then deduce results for the passage times. (Joint work with Victor Rivero, CIMAT, Mexico)

**Monday**, 4th February 2013, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Dr. David Belius (ETH Zürich, Switzerland)

Title: Gumbel fluctuations of certain cover times

Abstract: The cover time is the first time random walk on a graph has visited every vertex of that graph. Originating in computer science, cover times have been studied intensively by probabilists over the past few decades. One is usually interested in the behaviour of the cover time of large graphs. In my talk I will present some recent results regarding the fluctuations of the cover time of some specific but important graphs. The fluctuations in these cases turn out to follow the Gumbel distribution (in the case of the d-dimensional discrete torus of dimension at least 3 this resolves a long standing folklore conjecture). The meaning of the result is that despite the strong correlations present, the cover time behaves as if each vertex of the graph is covered independently.

**Wednesday**, 6th February 2013, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Dr. Sebastian Andres (Universität Bonn, Germany)

Title: Invariance Principle for the Random Conductance Model with ergodic

unbounded Conductances

Abstract: In this talk we consider a continuous time random walk **X** on **Z**^{d} in an environment of random conductances taking values in [0,∞). Assuming that the law of the conductances is ergodic with respect to space shifts, we present a quenched invariance principle for **X** under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser's iteration scheme. (Joint work with J.-D. Deuschel and M. Slowik.)

**Monday**, 11th February 2013, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Robert Graf (LMU München, Germany)

Title: A forest-fire model on the upper half-plane

Abstract: We consider a discrete forest-fire model on the upper half-plane of the two-dimensional square lattice. Each site can have one of the following two states: "vacant" or "occupied by a tree". At the starting time all sites are vacant. Then the process is governed by the following random dynamics: Trees grow at rate 1, independently for all sites. If an occupied cluster reaches the boundary of the upper half-plane or if it is about to become infinite, the cluster is instantaneously destroyed, i.e. all of its sites turn vacant. Additionally, we demand that the model is invariant under translations along the x-axis. We prove that such a model exists and arises naturally as a subsequential limit of forest-fire processes in finite boxes when the box size tends to infinity. Moreover, the model exhibits a phase transition in the following sense: There exists a deterministic critical time such that before the critical time, only sites close to the boundary have been affected by destruction, whereas after the critical time, sites on the entire half-plane have been affected by destruction.

**Wednesday**, 13th February 2013, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Marion Wörndl (LMU München, Germany)

Title: Asymptotic behaviour of exit times of one-dimensional Random Walks

and then subsequently

Sophie Dürre (LMU München, Germany)

Title: The critical case of Lamperti problem

**Thursday**, 14th February 2013, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück

Dr. Oleg Szehr (Technische Universität München, Germany)

Title: Spectral convergence bounds for finite classical and quantum Markov chains

Abstract: In this talk I present a new framework that yields spectral bounds on norms of functions of transition maps for finite, homogeneous Markov chains. The techniques employed work for bounded semigroups, in particular for classical as well as for quantum Markov chains and they do not require additional assumptions like detailed balance, irreducibility or aperiodicity. I use the method in order to derive convergence bounds that improve significantly upon known spectral bounds. The core technical observation is that power-boundedness of transition maps of Markov chains enables a Wiener algebra functional calculus in order to upper bound any norm of any holomorphic function of the transition map.

**Thursday**, 21st March 2013, 16:15, LMU, room B 251, Theresienstr. 39, Munich

Jonas Kukla, Ludwig-Maximilians-Universität München

Title: The distribution of the maximum degree in random trees

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