# Colloquium in probability and other talks in winter term 2017/18

**Organisers**: Nina Gantert (TUM), Noam Berger (TUM), Markus Heydenreich (LMU), Franz Merkl (LMU), Silke Rolles (TUM), Konstantinos Panagiotou (LMU), Sabine Jansen (LMU),

**Talks**:

**Monday**, 18^{th} September 2017, 15:30, LMU, room B 252, Theresienstr. 39, Munich

Andrea Schmidbauer (LMU)

Title: Site percolation in high dimensions

Abstract: Percolation usually studies the random sub-lattice consisting of occupied bonds. In contrast to the majority of the literature, we study site percolation and restrict attention to the hypercubic lattice Zd, where each site is occupied with probability p 2 [0; 1]. A key result in high-dimensional percolation is the so-called infrared bound that has been proven by Hara and Slade in 1990 for bond percolation. The aim of this presentation is to explain how the infrared bound can be proven for site percolation. The proof makes use of a combinatorial expansion technique, the so-called percolation lace-expansion. Our lace-expansion analysis covers the nearest-neighbor model.

**Monday**, 18^{th} September 2017, 16:30, LMU, room B 252, Theresienstr. 39, Munich

Leonid Kolesnikov (LMU)

Title: The critical 1-arm exponent for the Ising model on Cayley trees

Abstract: The ferromagnetic Ising model is one of the most extensively studied models from statistical mechanics. Here, we consider a regular tree as the underlying graph. We consider subtrees of depth n with fixed plus-valued boundary spins and investigate the expected spin value of the root of the tree with respect to the Gibbs measure on the subtree. It is well known that in the absence of an external field, the model undergoes a phase transition. At critical temperature, the influence of a plus boundary condition fades away when we take the limit in the distance between the root and the boundary of the subtree, i.e. the expected root spin is converging towards zero for n to infinity. Our main goal is to quantify the rate of this convergence.

A short introduction to the Ising model will be given at the beginning of the talk.

**Monday**, 23^{rd} October 2017, 14:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Katja Miller (TUM)

Title: Random walks on oriented percolation and in recurrent random environments

**Wednesday**, 25^{th} October 2017, 13:15, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Ercan Sönmez (Heinrich-Heine-Universität, Düsseldorf)

Title: Hausdorff dimension of multivariate operator-self-similar random fields

Abstract: The notion of Hausdorff dimension has been introduced in order to characterize sets which do possess a fractional pattern, commonly referred to as fractals. A typical feature of fractals is that they exhibit reappearing patterns, i.e. many fine details of the set resemble the whole set, a phenomenon which is called self- similarity. In case of multivariate self-similar random fields self-similarity means that a time-scaling corresponds statistically to a scaling in the state space, where the scaling relation is with respect to suitable matrices. This talk provides the first results on the sample paths and fractal dimensions of such fields, including quite general scaling matrices. A short introduction to the notion of Hausdorff dimension will also be given.

**Thursday**, 2^{th} November 2017, 14:30,TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Stein Andreas Bethuelsen (TUM)

Title: On the projection of the two-dimensional Ising model onto a line

Abstract: Consider the plus-phase of the two-dimensional Ising model below the critical temperature. Schonmann (1988) proved that the projection of this measure onto a line is not a Gibbs measure. In this talk I will review this result, as well as some more recent advances, and discuss some open questions related to Schonmanns work. In particular, I will focus on the following question: is the projection considered by Schonmann one-sided Gibbs? (in order words, is it a g-measure?) Based on joint work with Diana Conache (TUM).

**Monday**, 6^{th} November 2017, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Prof Simone Cerreia Vioglio (Universität Bocconi, Mailand)

Title: Choquet Integration of Self-adjoint Operators

Abstract: Comonotonicity and comonotonic additivity are at the base of the theory of Choquet integration which has many applications in Economics and Statistics. The main contribution of this work is to propose a definition of comonotonicity for elements of B(H), i.e. bounded self-adjoint operators defined over a complex Hilbert space H. We show that comonotonicity coincides with a form of commutativity. We also de
fine the notion of Choquet expectation for elements of B(H): a natural generalization of quantum expectations. We characterize Choquet expectations as the real-valued functionals over B(H) which are comonotonic additive, c-monotone, and normalized.

**Monday**, 13^{th} November 2017, 16:30,TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Dmitry Zaporozhets, (St. Petersburg Department of Steklov Mathematical Institute, St. Petersburg)

Title: Random convex hulls

Abstract: A basic object of Stochastic geometry is a random convex polytope. We will discuss several models including convex hulls of random walks and Gaussian polytopes.

**Monday**, 27^{th} November 2017, 16:30, LMU, room B 252, Theresienstr. 39, Munich

Dr. Timo Hirscher (Universität Stockholm)

Title: Consensus formation in the Deffuant model

Abstract: n 2000, Deffuant et al. introduced an interaction scheme to model opinion formation in large groups: Given a network graph and initial opinions, neighbors interact pairwise and either approach a compromise if their disagreement is below a given threshold or ignore each other if not. Concerning the asymptotics of the model, one central question is whether the whole group achieves a general consensus in the long run or splits into irreconcilable parts. We studied this model featuring univariate opinions on integer lattices and infinite percolation clusters as underlying interaction networks. The generalization of the original model to multivariate and measure-valued opinions was analyzed on the simple network given by the doubly-infinite path Z.

**Monday**, 11^{th} December 2017, 16:00, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Jan Nagel (Technische Universiteit Eindhoven)

Title: Random walk on a barely supercritical branching random walk

Abstract: The motivating question for this project is how a random walk behaves on a barely supercritical percolation cluster, that is, an infinite percolation cluster when the percolation probability is close to the critical value. As a more tractable model, we approximate the percolation cluster by the embedding of a Galton-Watson tree into the lattice. When the random walk runs on the tree, the embedded process is a random walk on a branching random walk. Now we can consider a barely supercritical branching process conditioned on survival, with survival probability approaching zero. In this setting the tree structure allows a fine analysis of the random walk and we can prove a scaling limit for the embedded process under a nonstandard scaling. The talk is based on a joint work with Remco van der Hofstad and Tim Hulshof.

**Monday**, 11^{th} December 2017, 17:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr Daniel Ueltschi (University of Warwick)

Title: Random interchange model on the complete graph and the Poisson-Dirichlet distribution

Abstract: In 2005, Schramm considered the random interchange model on the complete graph and he proved that the lengths of long cycles have Poisson-Dirichlet distribution PD(1). If one adds the weight 2^{#cycles}, one gets Toth's representation of the quantum Heisenberg model. In this case, we prove (essentially) that long cycles have distribution PD(2). In a related model of random loops, that involves "double bars" as well as "crosses", we prove that long loops have distribution PD(1). Joint work with J. Björnberg and J. Fröhlich.

**Monday**, 18^{th} December 2017, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Caio Teodoro De Magalhaes Alves (Universität Leipzig)

Title: Random graphs generated by preferential attachment rules with edge-steps

Abstract: In this talk I intend to present my recent work with Rodrigo Ribeiro (IMPA) and professor Rémy Sanchis (Federal University of Minas Gerais) regarding graphs generated by a preferential attachment rule that allows connections between already existent vertices to be formed at each new step according to some probability (that may be time dependent). We show, among other things, that the resulting graph has asymptotically almost surely a large complete subgraph of polynomial order when the density of new edges is at least constant. We also study the model when the probability of adding a new vertex converges to zero, and show the different behaviors it exhibits for different speeds of convergence.

**Monday**, 22^{th} January 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Dr. Tobias Hurth (Université de Neuchâtel)

Title: Invariant densities for randomly switched ODEs

Abstract: We consider random dynamical systems characterized by Poissonian switching between finitely many deterministic vector fields on a smooth manifold. Under a hypoellipticity condition at an accessible point on the manifold, the invariant measure of the associated Markov semigroup, if it exists, is unique and absolutely continuous. Unlike in the case of hypoelliptic diffusions, the density of the invariant measure need not be globally smooth due to possible contraction at critical points. Understanding the behavior of the density is then a challenging problem. The talk aims to give an introduction to the ergodic theory of systems with random switching. It is based on work with Yuri Bakhtin, Sean Lawley and Jonathan Mattingly.

**Monday**, 29^{th} January 2018, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Yulia Petrova (St. Petersburg)

Title: Exact $L_2$-small ball probabilities for finite-dimensional perturbations of Gaussian processes: spectral method.

Abstract: I consider the problem of small ball probabilities for Gaussian processes in $L_2$-norm. I focus on the processes which are important in statistics (e.g. Kac-Kiefer-Wolfowitz processes), which are finite dimentional perturbations of Gaussian processes. Depending on the properties of the kernel and perturbation matrix I consider two cases: non-critical and critical. For non-critical case I prove the general theorem for precise asymptotics of small deviations. For a huge class of critical processes I prove a general theorem in the same spirit as for non-critical processes, but technically much more difficult. At the same time a lot of processes naturally appearing in statistics (e.g. Durbin, detrended processes) are not covered by those general theorems, so I treat them separately using methods of spectral theory and complex analysis. First I will give an introduction to the topic and a historical review, and then I will tell about my results.

**Thursday**, 1^{th} February 2018, 14:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Carina Geldhauser (St. Peterburg)

Title: Invariant measures of the Euler equation in vorticity form

Abstract: When investigating the turbulent behaviour in a flow, it seems natural to study invariant measures of the Euler equation. In this talk we desribe the system of point vortices, derived by Onsager from the Euler equation, and their associated Gibbs measures. In case of constant circulations, Caglioti, Lions, Marchioro and Pulvirenti showed that the Gibbs measures concentrate on very particular stationary solutions of the 2D Euler equation in the weak limit and they satisfy a variational principle. Furthermore, we discuss available results and open problems for systems of point vortices with random circulations on the 2D Torus.

**Monday**, 5^{th} February 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Dr. Christian Hirsch (LMU)

Title: Continuum percolation for Cox point processes

Abstract: We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of non-trivial sub- and super-critical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in large-radius, high-density and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives.

**Thursday**, 8^{th} February 2018, 14:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dominik Schmid (TUM)

Title: Mixing times for the exclusion process

Abstract: The exclusion process is one of the best-studied examples of an interacting particle system. In this talk, we focus on the exclusion process on finite graphs. Our goal is to study the total variation mixing time, which is the standard way of measuring the speed of convergence to equilibrium. We give an overview over some recent results on the mixing time of the exclusion process on the line segment as well as a selection of open problems in this area.

**Mittwoch**, 14^{th} March 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Prof. Dr. Vincent Tassion (ETH Zürich)

Title: The phase transition for Boolean percolation

Abstract: Based on joint works with Daniel Albergh, Hugo Duminil-Copin, Aran Raoufi and Augusto Teixeira. Consider Poisson-Boolean percolation in R^d. Around every point of a Poisson point process of intensity lambda, draw independently a ball with random radii. I will discuss the sharpness of the phase transition in lambda for this process. In particular, I will present sharp bounds for the cluster of the origin to have radius greater than n in the subcritical regime, and how these bounds depend on the radii distributions. >

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