# Oberseminar Wahrscheinlichkeitstheorie und andere Vorträge "TUM-LMU" im Sommersemester 2016

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

**Talks**:

**Monday**, 18^{th} April 2016, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Dr. Christian Hirsch (WIAS Berlin)

Title: On maximal hard-core thinnings of stationary particle processes

Abstract: We consider existence and uniqueness of subclasses ofstationary hard-core particle systems arising as thinnings of stationaryparticle processes. These subclasses are defined by natural maximalitycriteria. We investigate two specific criteria, one related to theintensity of the hard-core particle process, the other one being a localoptimality criterion. In fact, the criteria are equivalent undersuitable moment conditions. We show that stationary hard-core thinningssatisfying such criteria exist and are frequently unique. Moreprecisely, uniqueness holds in subcritical and barely supercriticalregimes of continuum percolation. Additionally, based on the analysis ofa specific example, we argue that fluctuations in grain sizes can playan important role for establishing uniqueness at high intensities. Thistalk is based on joint work with Günter Last."

**Monday**, 9^{th} May 2016, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Prof. Dr. Anja Sturm (Universität Göttingen)

Title: Interacting particle systems with cooperative branching and coalescence

Abstract: In this talk, we focus on a one-dimensional model of individuals/particles performing independent random walks on Z in which only pairs of individuals can produce offspring (cooperative branching) and individuals that land on an occupied site merge with the individual present on that site (coalescence). In a biological context, the resulting cooperative branching-coalescent describes a simple population dynamics with reproducing pairs of particles. Coalescence models death due to competition for resources. We argue that the model can also be used as an approximation to a population with two sexes in which only pairs of the opposite sex can reproduce. In addition, the process also describes the interface dynamics of a multi type voter model in which rare types have an advantage. Mathematically, the cooperative branching-coalescent has interesting properties: We show that the system undergoes a phase transition as the branching rate is increased. For small branching rates the upper invariant law is trivial and the process started with finitely many individuals a.s. ends up with a single individual. Both statements are not true for high branching rates. We also study the decay of the population density of the process started in the fully occupied state if the branching rate is small enough. This talk is based on joint work with Jan Swart (UTIA Prague).

**Monday**, 23th May 2016, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Verena Reinl and Christine Birkmüller (LMU)

Title:Upper and lower bounds for l-colored graphs

Abstract: This talk and the underlying Master's theses discuss properties of l-colored graphs, i.e.of graphs equipped with a distinct l-coloring identied with its corresponding partitionof the vertex set into color-classes. Our main goal is to study the expected sizes of thesecolor-classes of a graph on n vertices drawn uniformly at random from the set of alll-colored graphs.Instead of working with the sizes n_1;...; n_l we will use its deviations d_i = n_i-floor{n/l},combined to a deviation vector d. Our first interesting result is then that, with someassumptions on l, we get d^Td < 18l with high propability. We use this to prove anupper resp. a lower bound for the maximal deviation with respect to its absolute value.We will see that for large n we can expect the maximum to be of order sqrt{2log l}.Finally we will use a similar argument to prove that for a given deviation s with |s|<f_1,where f_1 denotes the upper bound for the maximal deviation, the number of color-classesof size floor{n/l}+s can be found with high probability in an interval [g_s*l; f_s*l] where both g_sand f_s are constants depending solely on s and have the form 2^{1/2*s^2+O(s)}.

**Monday**, 30^{th} May 2016, 16:00 and 17:30, LMU, room B 251, Theresienstr. 39, Munich

Prof. Dr. Gordon Slade (UBC Vancouver, Kanada)

Title: Critical exponents for long-range O(n) modelsAbstract: We consider the critical behaviour of long-range O(n) models for n greater than or equal to 0. For n=1,2,3,... these are phi^4 spin models. For n=0 it is the weakly self-avoiding walk. We prove existence of critical exponents for the susceptibility and the specific heat, below the upper critical dimension. This is a rigorous version of the epsilon expansion in physics. The proof is based on a rigorous renormalisation group method developed in previous work with Bauerschmidt and Brydges.

Prof. Dr. Volker Betz (TU Darmstadt)

Title: Spatial Random Permutations

Spatial random permutations are implemented by probability measures on permutations of a set with spatial structure; these measures are made so that they favor permutationsthat map points to nearby points. The strength of this effect is encoded in a parameter alpha > 0, where larger alpha means stronger bias toward short jumps. I will introduce some variants of the model, and explain the connections to the theory of Bose-Einstein condensation. Then I will present a few older results, as well as very recent progress made jointly with Lorenzo Taggi (TU Darmstadt) for the regime of large alpha. Finally, I will discuss two conjectures suggested by numerical simulation: in two dimensions, the model appears to exhibit a Kosterlitz-Thouless phase transition, and there are reasons to believe that in the phase of algebraic decay of correlations, long cycles are Schramm-Löwner curves, with parameter between 4 and 8 depending on alpha.

**Monday**, 6^{th} June 2016, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Prof. Dr. Piotr Milos (University of Warsaw, Poland)

Title: Extremal individuals in branching systems

Abstract: Branching processes have been subject of intense and fascinating studies for a long time. In my talk I will present two problems in order to highlight their rich structure and various technical approaches in the intersection of probability and analysis. Firstly, I will present results concerning a branching random walk with the time-inhomogeneous branching law. We consider a system of particles, which at the end of each time unit produce offspring randomly and independently. The branching law, determining the number and locations of the offspring is the same for all particles in a given generation. Until recently, a common assumption was that the branching law does not change over time. In a pioneering work, Fang and Zeitouni (2010) considered a process with two macroscopic time intervals with different branching laws. In my talk I will present the results when the branching law varies at mesoscopic and microscopic scales. In arguably the most interesting case, when the branching law is sampled randomly for every step, I will present a quenched result with detailed asymptotics of the maximal particle. Interestingly, the disorder has a slowing-down effect manifesting itself on the log level. Secondly, I will turn to the classical branching Brownian motion. Let us assume that particles move according to a Brownian motion with drift \mu and split with intensity 1. It is well-know that for \mu\geq \sqrt{2} the system escapes to infinity, thus the overall minimum is well-defined. In order to understand it better, we modify the process such that the particles are absorbed at position 0. I will present the results concerning the law of the number of absorbed particles N. In particular I will concentrate on P(N=0) and the maximal exponential moment of N. This reveals new deep connections with the FKPP equation. Finally, I will also consider -\sqrt{2}<\mu<\sqrt{2} and N_t^x the number of particles absorbed until the time t when the system starts from x. In this case I will show the convergence to the traveling wave solution of the FKPP equation for an appropriate choice of x,t->\infty. The results were obtained jointly with B. Mallein and with J. Berestycki, E. Brunet and S. Harris respectively.

**Wednesday**, 08^{th} June 2016, 14:00, TUM, room 2.01.11, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Eviatar Procaccia (Texas University)

Title: Can one hear the shape of a random walk?

Abstract: We consider a Gibbs distribution over random walk paths on the square lattice, proportional to a random weight of the path’s boundary . We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plain to the sum of their principle eigenvalue and perimeter (with respect to the first passage percolation norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes.

**Monday**, 13^{th} June 2016, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. David Mason (University of Delaware)

Title: Stochastic Compactness of Levy Processes

Abstract: Siehe Link.

**Monday**, 20^{th} June 2016, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Satoshi Handa (Hokkaido University,Japan)

Title: Mean-field bound on the 1-arm exponent for Ising model in high dimensions

Abstract: Arm exponents characterize critical behaviour of spatial models in Statistical Mechanics. In this talk I present results concerning the 1-arm exponent for the Ising model in high dimensions. I shall give a comparison with the 1-arm exponent for percolation, which has been obtained recently. Subsequently, I explain our approach for the mean-field bound in the Ising model.

**Monday**, 04^{th} July 2016, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Alexander Mendelsohn (LMU)

Title: The vectorial Deffuant model

Abstract: The vectorial Deffuant model is an extension of the voter model. It is a multi-state spin system which aims to model opinion dynamics or cultural dissemination and accounts for “homophily” with a so called ‘threshold for interaction’. We consider the vectorial Deffuant model on the 1-dimensional lattice and investigate its limiting behaviour. Three classes of behaviour are introduced: * clustering - complete consensus is achieved in the limit * fixation - a limiting (random) configuration is achieved eventually * fluctuation - configuration changes at arbitrarily large times The central objective is to find relations between the size of the state space and threshold parameters and the limiting behaviour of the model.

**Freitag**, 08^{th} July 2016, 10:15, LMU, room B 252, Theresienstr. 39, Munich

Dr. Sebastian Ziesche (Karlsruhe Insitute of Technology)

Title: The Ornstein-Zernike equation for the pair-connectedness function of stationary cluster processes

Abstract: In a seminal paper Ornstein and Zernike proposed in 1914 to split the interaction between molecules in a liquid into a direct and an indirect part. While the resulting spatial convolution equation is of great important in physics, it seems to be hardly known among mathematicians. In the first part of this talk we consider the pair-connectedness function (PCD) of a rather general stationary cluster model. Combining point process methods with analytic tools for solving integral equations we show that the associated Ornstein-Zernike equation (OZE) admits a unique solution in the whole subcritical regime. In the second part of the talk we consider the special case of a Poisson Boolean model with deterministic grains and show that the solution of the OZE is an analytic function of the intensity. Moreover, for small intensities there is a simple combinatorial way (based on the concept of pivotal diagrams) to express the coefficients of the power series in terms of the corresponding coefficients of the PCD. In the final part of the talk we shall briefly discuss the random connection model and propose some directions for future research. This talk is based on joint work with Günter Last (Karlsruhe).

**Freitag**, 08th July 2016, 11:00, LMU, room B 252, Theresienstr. 39, Munich

Prof. Dr. Günter Last (Karlsruhe Institute of Technology)

Title: Exact and asymptotic results for the Poisson random connection model

Abstract: In the first part of the talk we consider a Poisson process on a general phase space. The random connection model is obtained by connecting two Poisson points according to some random rule which is independent for different pairs. We shall discuss first and second order properties of the point processes counting the clusters of a given size. In the second part of the talk we specialize to an Euclidean phase space and prove a central limit theorem for the number of clusters in a growing observation window. The proof is based on some new Berry-Esseen bounds for the normal approximation of functionals of a pairwise marked Poisson process. This talk is based on joint work with Franz Nestmann (Karlsruhe) and Matthias Schulte (Bern).

**Monday**, 11^{th} July 2016, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Dr. Jan Nagel (TUM)

Title: Large deviations for spectral measures and sum rules

Abstract: We show a large deviation principle for the weighted spectral measure of random matrices corresponding to a general potential. Unlike for the empirical eigenvalue distribution, the speed reduces to n and the rate function contains a contribution of eigenvalues outside of the limit support. As an application, we show how this large deviation principle yields a probabilistic proof of the celebrated Killip-Simon sum rule: a remarkable relation between the entries of a Jacobi-operator and its spectral measure. We also obtain new variants of such sum rules. This talk is based on a joint work with Fabrice Gamboa and Alain Rouault.

**Monday**, 18^{th} July 2016, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Sebastian Bordt

Title: Cutoff for Random walks on random graphs

Abstract: Random walk on the giant component of the Erdős–Rényi random graph exhibits cutoff, which is fast convergence to the stationary distribution within a specific time frame, the so-called cutoff window. The same is true for random walk on d-regular graphs, and more generally, random walks on random graphs with given degree sequences. This is in contrast to random walk on other classes of graphs, for example ladder graphs or cycles. The talk is based on my Master's thesis and aims at explaining why cutoff occurs for some classes of graphs and not for others, and how to prove that. I will specifically discuss coupling techniques. The theory will be illustrated with numerical simulations.