# Oberseminar Wahrscheinlichkeitstheorie und andere Vorträge im Wintersemester 2018/19

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

**Talks**:

**Monday**, 17^{th} September 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Anne-Marie Mößnang (LMU, MSc presentation)

Title: Sharp phase transition for confetti percolation

Abstract: Recently, Duminil-Copin, Raoufi and Tassion developed a new method to prove sharp phase transition for Voronoi percolation even in higher dimensions. The idea is based on two main steps: For $S_n(0) := \{x \in \mathbb{R}^d : ||x|| = n\}$ and $\theta_n(p) := P_p(0 \leftrightarrow S_n(0))$, they first prove a family of differential inequalities regarding $\theta_n(p)$. Here, they make use of a randomized algorithm, which determines the function $f := 1_{0 \leftrightarrow S_n(0)}$, and of the OSSS inequality, to estimate the variance of $f$. Second they employ a Lemma to $\theta_n(p)$, which verifies the sharp phase transition. In the talk we transfer this method to prove sharp phase transition for confetti percolation in $\mathbb{R}^d \times (- \infty, 0]$.**Im Anschluss daran** (ca. 17:15): Florian Rudiger (LMU, MSc presentation)

Title:Recurrence and transience of random geometric graphs

Abstract: In this talk, we prove for various graphs that the random walk is recurrent or transient. While in one case the random walk almost surely visits every vertex of the graph infinitely many times, in the other case it eventually escapes any finite set of vertices and never returns. Under certain assumptions on the underlying point process, we apply results from Gurel-Gurevich, Nachmias and Rousselle to get recurrence results for graphs in the plane and transience results for higher dimensions. Apart from that we will mention some classes of point processes for which our results hold.

**Thursday**, 27^{th} September 2018, 15:00, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Rangel Baldasso (Bar Ilan University)

Title: Spread of an infection on the zero range process

Abstract: We consider the spread of an infection on top of a moving population. The environment evolves as a zero range process on the integer lattice starting in equilibrium. At time zero, the set of infected particles is composed by those which are on the negative axis, while particles at the right of the origin are considered healthy. A healthy particle immediately becomes infected if it shares a site with an infected particle. We prove that the front of the infection wave travels to the right with positive and finite velocity.

**Monday**, 15^{th} October 2018, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Benedikt Stufler (University of Zurich)

Title: Invariance principles for random planar structures

Abstract: Invariance principles provide a universal description of the behaviour of a general class of random objects. For example, if a random walk lies in the domain of attraction of a stable law, then it converges after an appropriate rescaling to the corresponding stable Lévy process. The past decades have seen rapidly growing research activity on related universal limit objects for random planar structures, such as trees or graphs embedded on a fixed surface. The talk is meant to give an introduction to this topic, outline some selected results, and discuss future research directions.

**Monday**, 22^{th} October 2018, 15:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Tal Orenshtein (Humboldt-Universität zu Berlin)

Title: Ballistic RWRE as rough paths - convergence and area anomaly

Abstract: We shall discuss our work on ballistic RWRE. We show that the annealed functional CLT holds in the rough path topology, which is stronger than the uniform one. This yields an interesting phenomenon: the scaling limit of the area process is not solely the Levy area, but there is also an additive linear correction which is called the area anomaly when is non-zero. Moreover, the latter is identified in terms of the walk on a regeneration interval and the asymptotic speed. A general motivation for achieving limit theorems for discrete processes in the rough path topology is the following property, which might be useful e.g., for simulations. Consider a nice difference equation driven by the recentered walk. A result by D. Kelly gives a scaling limit to the corresponding SDE, with an appropriate correction expressed explicitly in terms of the area anomaly. This is a joint work in progress with Olga Lopusanschi (Paris-Sorbonne)

**Monday**, 22^{th} October 2018, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Timo Hirscher (Stockholm University)

Title: The Schelling model for segregation on Z

Abstract: In 1969, economist T. Schelling invented a simple model of interacting particles to explain racial segregation in American cities: The nodes of a simple graph are occupied by agents of different kinds and each of them is inclined to have neighbors of its own kind. While Schelling used pennies and dimes on a checkerboard to implement some old-school-simulations on a finite instance, we are interested in the corresponding model on Z, the one-dimensional integer lattice. It turns out that the asymptotics are similar to the one of the voter model - but only if the range of a move is unbounded.

**Monday**, 29^{th} October 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Lukasz Zwonek (MSc presentation)

Title: "Duality of Markov Processes"

Abstract: We consider the duality of Markov processes with additional Feynman-Kac corrections. After introducing some basic examples from population genetics, we provide sufficient and necessary conditions for the existence of Feynman-Kac dual Markov chains in discrete time and finite spaces. The criteria will be formulated in a functional analytics fashion following the research of S. Jansen and N.Kurt.

**Monday**, 12^{th} November 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Kilian Matzke (LMU)

Title: The Random Connection Model at Criticality

Abstract: We consider the random connection model, which is a continuum percolation model. After introducing the model along with some basic tools, we adapt the lace expansion to the framework of the underlying continuum space Poisson point process. This allows us to derive the triangle condition above the upper critical dimension and furthermore to establish the infra-red bound. From this, mean-field behavior of the model can be deduced.

**Monday**, 19^{th} November 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Stefan Adams (University of Warwick)

Title TBA

**Tuesday**, 27^{th} November 2018, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Nicos Georgiou (University of Sussex)

Title: Last passage times in discontinuous environments.

Abstract: We are studying a last passage percolation model on the two dimensional lattice, where the environment is a field of independent random exponential weights with different parameters. Each variable is associated with a lattice vertex and its parameter is selected according to a discretization of lower semi-continuous parameter function that may admit discontinuities on a set of curves. We prove a law of large numbers for the sequence of last passage times, defined as the maximum sum of weights which a directed path can collect from (0, 0) to a target point (Nx, Ny) as N tends to infinity and the mesh of the discretisation of the parameter function tends to 0 as 1/N. The LLN is cast in the form of a variational formula, optimised over a given set of macroscopic paths. Properties of maximizers to the variational formula above are investigated in two models where the parameter function allows for analytical tractability. This is joint work with Federico Ciech.

**Thursday**, 13^{th} December 2018, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Christoph Thäle (Ruhr-Universität Bochum)

Title: TBA

**Monday**, 17^{th} December 2018, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Mykhaylo Shkolnikov (Princeton University)

Title: TBA

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