# Colloquium in probability and other talks in winter term 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.

**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.

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

Mykhaylo Shkolnikov (Princeton University )

Title: The supercooled Stefan problem

Abstract: The Stefan problem arising from the physics of supercooled liquids poses major mathematical challenges due to the presence of blow-ups, including even the definition of solutions. I will explain how the problem can be reformulated in probabilistic terms and how related particle system models lead to an appropriate notion of a solution. The solutions can be then studied by probabilistic techniques and a sharp description of the blow-ups can be established. Based on joint works with Sergey Nadtochiy and an ongoing joint work with Francois Delarue and Sergey Nadtochiy.

**Tuesday**, 18^{th} December 2018, 16:00, TUM, room MI 03.10.011, Boltzmannstr. 3, 85748 Garching (Technische Universität München)

Izabella Stuhl

Title:Hard-core model on 2D lattices

Abstract: It is well-known that in R2 the maximum-density conguration of hard-core (non-overlapping) disks of diameter D is given by a triangular/hexagonal arrangement (Fejes Tóth, Hsiang). If the disk centers are placed at sites of a lattice, say, a unit triangular lattice L2 or a unit square lattice Z2, then we get a discrete analog of this problem, with the Euclidean exclusion distance. I will discuss high-density Gibbs/DLR measures for the hard-core model on L2 and Z2 for a large value of fugacity z. According to the Pirogov-Sinai theory, the extreme Gibbs measures are obtained via a polymer expansion from domi- nating ground states. For the hard-core model the ground states are associated with maximally dense sublattices, and dominance is determined by counting defects in local excitations. On L2 we have a complete description of the extreme Gibbs measures for a large z and any D; a convenient tool here is the Eisenstein integer ring. For Z2, the situation is made more complicated by various (related) phenomena: sliding, non-tessellation etc. Here, some results are available; conjectures of various generality can also be proposed. A number of our results are computerassisted. This is a joint work with A. Mazel and Y. Suhov.

**Thursday**, 20^{th} December 2018, 16:30, LMU, room B349, Theresienstr. 37, Munich

Lutz Warnke (Georgia Tech, Atlanta)

Title: A dynamic view on the probabilistic method: random graph processes

Abstract: Random graphs are the basic mathematical models for large-scale disordered networks in many different fields (e.g., physics, biology, sociology). Since many real world networks evolve over time, it is natural to study various random graph processes which arise by adding edges (or vertices) step-by-step in some random way. The analysis of such random processes typically brings together tools and techniques from seemingly different areas (combinatorial enumeration, differential equations, discrete martingales, branching processes, etc), with connections to the analysis of randomized algorithms. Furthermore, such processes provide a systematic way to construct graphs with "surprising" properties, leading to some of the best known bounds in extremal combinatorics (Ramsey and Turan Theory). In this talk I shall survey several random graph processes of interest, and give a glimpse of their analysis. In particular, if time permits, we shall illustrate one of the main proof techniques (the "differential equation method") using a simple toy example.

**Monday**, 14^{th} January 2019, 16:30, LMU, room B252, Theresienstr. 39, Munich

Volker Betz (TU Darmstadt)

Title: Scaling limit of a self-avoiding walk interacting with spatial random permutatinos

Abstract: We consider a self-avoiding walk conditioned to cross a large box and weighted with an energy proportional to the number of steps it takes to do so. We embed this walk into a background of self-avoiding nearest-neighbour loops or, equivalently, nearest-neighbour spatial random permutations.

**Monday**, 21^{th} January 2019, 16:30, LMU, room B252, Theresienstr. 39, Munich

Alexander Drewitz (University of Cologne)

Title: Geometry of Gaussian free field sign clusters and random interlacements

Abstract: We consider two fundamental percolation models with long-range correlations: Level set percolation for the Gaussian free field (GFF) and percolation of the vacant set of random interlacements. Both models have been the subject of intensive research during the last decades. In this talk we focus on structural properties of the level set percolation of the GFF. In particular, we establish the non-triviality of a phase in which two infinite sign clusters dominate, and their complement only has small connected components. While the respective results are new for $\Z^d$ as the underlying graph, we also cover more intricate geometries such as transient graphs with subdiffusive random walk behavior. As a consequence, we answer an open problem on the non-triviality of the phase transition of the vacant set of Random Interlacements on such geometries.

**Monday**, 28^{th} January 2019, 16:30, LMU, room B252, Theresienstr. 39, Munich

Fabio Frommer (Johannes-Gutenberg Universität Mainz)

Title: The Henderson problem

Abstract: The inverse Henderson problem of statistical mechanics concerns classical particles in continuous space which interact according to a pair potential depending on the distance of the particles. Roughly stated, it asks for the interaction potential given the equilibrium pair correlation function of the system. In 1974 Henderson proved that this potential is uniquely determined in a canonical ensemble and he claimed the same result for the thermodynamical limit of the physical system. Here we provide a rigorous proof of a slightly more general version of the latter statement using Georgii's version of the Gibbs variational principle.

**Monday**, 4^{th} February 2019, 16:30, LMU, room B252, Theresienstr. 39, Munich

Thomas Beekenkamp (LMU)

Title: Sharpness of the Percolation Phase Transition for the Contact Process on Z^d

Abstract: The contact process is a model for the spread of infection on a graph. Each vertex of Z^d is either healthy or infected. Infected vertices become healthy with rate 1, independent of the rest of the process. Healthy sites become infected with a rate of lambda times the number of neighbours that are infected, where lambda is the parameter of the model. There exists a critical value for lambda above which there exists an infinite cluster and below which all clusters are finite. We show that this phase transition is sharp, i.e., below this critical value all clusters are exponentially small. The proof draws on a series of recent papers by Duminil-Copin, Raoufi and Tassion in which they prove sharp phase transitions for a variety of models using the OSSS inequality.

**Wednesday**, 27^{th} February 2019, 10:30, LMU, room B252, Theresienstr. 39, Munich

Theresa Ullmann (LMU, MSc presentation)

Title: Hawkes Processes on Networks

Abstract: How do neurons in the brain interact with each other? This is a question not just for neuroscientists, but also for mathematicians, as mathematical models can help to elucidate some phenomena of neural networks. We give an example for such a model, namely Hawkes processes on networks. The set of neurons and their connections form a network, and each neuron in this network is assigned a Hawkes process (a special point process) that models its spike train. We outline a proof strategy for existence and uniqueness of Hawkes processes on networks, where we follow the work of Delattre/Fournier/Hoffmann 2016. We then discuss the large-time behaviour of the Hawkes processes, where we extend the results of Delattre et al. to more general networks.**Im Anschluss daran **um 11:30,

Johannes Laub (LMU, MSc presentation)

Title: Survival and extinction of the contact process on a symmetric graph

Abstract: We give a construction of a symmetric graph, on which the contact process survives on the whole graph, but dies out if you remove three privileged edges. For this, we will first introduce the graphical construction of the contact process on graphs, give the construction of our graph and the proof of the above statement.

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