# Oberseminar Wahrscheinlichkeitstheorie und andere Vorträge im Wintersemester 2015/16

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

**Talks**:

**Monday**, 19^{th} October 2015, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Dr. Florian Völlering (TU Berlin)

Title: On the Symbiotic Branching Model

Abstract: The symbiotic branching model describes two interacting spatial populations whose evolution is given by a system of correlated SPDEs. I will first introduce the model and its basic features, which include a competition between dispersal and smoothening of population densities by migration and and inter-species competition due to the symbiotic branching. Macroscopically the populations become separated, which we study in more detail by sending the branching rate to infinity. In the special case of perfectly anticorrelated noise, which is related to the stepping stone model, we will fully describe the solution, which is given by a system of annihilating Brownian motions. Joint work Marcel Ortgiese (WWU Münster) and Matthias Hammer (TU Berlin).

**Monday**, 26^{th} October 2015, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Ron Rosenthal (ETH Zürich)

Title: Simplicial branching random walks

Abstract: We will discuss a new stochastic process on general simplicial complexes which allows to study their spectral and homological properties. Some results for random walks on graphs will be shown to hold in this general setting. In addition, we will discuss transience/recurrence classification for the process on high-dimensional analogues of regular trees, and show how to construct solutions to the high-dimensional Dirichlet problem.

**Monday**, 02^{th} November 2015, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Joost Jorritsma (Eindhoven University of Technology)

Title: Scale-free percolation

Abstract: Scale-free percolation is a percolation model on the d-dimensional lattice introduced by Deijfen, van der Hofstad and Hooghiemstra. Each vertex x has a random weight W_x, such that the weights are i.i.d. Pareto-distributed random variables with parameter tau-1. Conditionally on the weights, the edges in the graph are independent. Two vertices x and y are connected by an edge with probability p_{xy}=1-e^{-lambda W_x W_y / |x-y|^alpha}, where lambda>0 is called the percolation parameter and alpha>0 the long-range parameter. One of the crucial parameters in this model turns out to be gamma := alpha (tau-1) / d. I present basic properties of the model, such as degree-sequences (power-law with parameter gamma), critical percolation values (transitions for gamma=2) and graph distances. Our new results concern boundedness of the graph-distance for gamma >= 1 and transience vs. recurrence transitions in random walks. In this study we continue with some questions that initially were left open, such as boundedness of the graph-distance when gamma >= 1 and random walks on the percolation cluster. The model interpolates in some sense between the inhomogeneous random graphs (IRRG) and long-range percolation (LRP). We have the power-law distribution for degrees as in the IRRG, but also have more spatial structure as LRP. Real-life networks, such as social networks, internet and many others, have both properties. Trying to understand those networks is one of the reasons why studying this model and its phase-transitions are interesting.

**Monday**, 9^{th} November 2015, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Prof. Dr. Ewa Damek (Uniwersytet Wrocławski, Wrocław, Poland)

Title: Large deviation estimates for exceedance times of perpetuity sequences

Abstract: Click here.

**Monday**, 16^{th} November 2015, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Prof. Dr. Peter Mörters (University of Bath)

Title: Title: Robustness of spatial preferential attachment networks

Abstract: A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrarily small positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the unit circle and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent τ of the degree distribution and the exponent δ at which the connection probability decreases with the distance of two vertices. We show that the network is robust if τ < 2 + 1/δ , but fails to be robust if τ > 2 + 1/(δ−1) . This is the first instance of a scale-free network where robustness depends not only on its degree distribution but also on its clustering features. This is joint work with Emmanuel Jacob (ENS Lyon).

**Monday**, 23^{th} November 2015, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Marinus Gottschau (LMU)

Title: On the size of the infected set in the Bootstrap percolation process

Abstract: In this talk we focus on r-bootstrap percolation, which is a process on a graph where initially a set A_0 of vertices gets infected. Now subsequently, an uninfected vertex becomes infected if it is adjacent to at least r infected vertices. Call A_f the set of vertices that is infected after the process stops.First we present a theorem for degenerate graphs that bounds the size of the final infected set. More precisely for a d-degenerate graph, if r>d, we prove that the size of the set A_f is bounded from above by (1+d/(r-d))|A_0|.In the second part of the talk we focus on the process on inhomogeneous random graphs, where we choose the weights of the graph according to a power-law distribution with exponent beta in (1,2). We determine two functions a^+_c(n)=o(n) and a_c(n)=o(n), which agree for certain parameter choices, such that if we choose A_0 of size a(n) randomly, then if a(n)<< a_c(n), a.a.s. the process does not evolve at all, but if a(n)>> a^+_c(n), we have that |A_f|>epsilon n for some epsilon >0 a.a.s.

**Monday**, 23^{th} November 2015, 17:15, LMU, room B 251, Theresienstr. 39, Munich

Kilian Matzke (LMU)

Title: The Saturation Time Of Graph Bootstrap Percolation

Abstract: The process of H-bootstrap percolation for a graph H is a cellular automaton, where, given a subset of the edges of K_n as initial set, an edge is added at time t if it is the only missing edge in a copy of H in the graph obtained through this process at time t-1. We discuss an extremal question about the time of K_r-bootstrap percolation, namely determining maximal times for an n-vertex graph before the process stops. We determine exact values for r=4 and find a lower bound for the asymptotics for r \geq 5 by giving an explicit construction.

**Monday**, 30^{th} November 2015, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Benedikt Stufler (LMU)

Title: local weak limits of random labelled graphs and outerplanar maps from classes satisfying a subcriticality condition

Abstract: A local weak limit of a sequence of random graphs is a random (possibly infinite) graph describing the asymptotic behaviour of the sequence locally around a fixed or randomly drawn root. We establish local weak limits of various classes of random structures, including random graphs from subcritical classes such as cacti graphs, outerplanar graphs, series parallel graphs, and random outerplanar maps from classes satisyfing a subcriticality condition such as unrestricted and bipartite outerplanar maps. We use combinatorial bijections (one classical and one recently established by the speaker in his work on the scaling limit of outerplanar maps with independent link weights) in order to apply results on Galton-Watson trees.

**Monday**, 7^{th} December 2015, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Prof. Dr. Jochen Blath (TU Berlin)

Title: Modeling the effect of individual dormancy in mathematical population genetics

Abstract: We investigate extensions of the classical Wright Fisher population model in which individuals can enter a reversible state of dormancy. Such dormant individuals create a seed bank that increases genetic variability of the population. We review several seed bank models and present scaling limits for type frequency processes and the corresponding genealogies. This naturally leads to stochastic delay differential equations and to new coalescent structures, where lineages are independently blocked from coalescence for extended periods of time. We provide some theoretical results for the long-time behaviour of these models, as well as predictions of patterns of genetic variability, which could be testet against real data.

**Monday**, 14^{th} December 2015, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

David Criens (TUM)

Title: Martingality in Terms of Semimartingale Problems

Abstract: Starting from the seventies mathematicians face the question whether a non-negative local martingale is a true martingale or a strict local martingale. In this article we answer this question from a semimartingale perspective. Our results are based on semimartingale problems, which were introduced in Jacod (1979). We derive sufficient and necessary conditions for a non-negative local martingale to be a true respectively a strict local martingale in terms of existence and local uniqueness properties of a semimartingale problem. As case studies we connect our result with the concepts of stochastic differential equations and classical local martingale problems. Moreover, in these two settings we state concrete conditions and give examples.

**Monday**, 21^{th} December 2015, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Mykhaylo Shkolnikov (Princeton University)

Title: Edge of beta ensembles and the stochastic Airy semigroup

Abstract: Beta ensembles arise naturally in random matrix theory as a family of point processes, indexed by a parameter beta, which interpolates between the eigenvalue processes of the Gaussian orthogonal, unitary and symplectic ensembles (GOE, GUE and GSE). It is known that, under appropriate scaling, the locations of the rightmost points in a beta ensemble converge to the so-called Airy(beta) process. However, very little information is available on the Airy(beta) process except when beta=2 (the GUE case). I will explain how one can write a distribution-determining family of observables for the Airy(beta) process in terms of a Brownian excursion and a Brownian motion. Along the way, I will introduce the semigroup generated by the stochastic Airy operator of Ramirez, Rider and Virag. Based on joint work with Vadim Gorin.

Gemeinsam mit dem Seminar: "Analysis und Zufall"

**Thursday**, 7^{th} January 2015, 16:30, LMU, room A 027 (Erdgeschoss), Theresienstr. 39, Munich

Prof. Dr. Konstantinos Panagiotou (LMU)

Title: Connectivity Thresholds in Random Graph Processes

Abstract: In an Achlioptas process, starting with a graph that has n vertices and no edge, in each round d edges are drawn uniformly at random, and using some rule exactly one of them is chosen and added to the evolving graph. For the class of Achlioptas processes we investigate how much impact the rule has on one of the most basic properties of a graph: connectivity. We study the prominent class of bounded size rules, which select the edge to add according to the component sizes of its vertices, treating all sizes larger than some constant equally. For such rules we provide a fine analysis that exposes the limiting distribution of the number of rounds until the graph gets connected, and we give a detailed picture of the dynamics of the formation of the single component from smaller components.

**Monday**, 18^{th} January 2016, 15:00, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Ph.D Tal Orenshtein (Institute Camille Jordan)

Title: Router walks on regular trees

Abstract: Following a joint work with Sebastian Mueller, we will discuss the problem of recurrence versus transience of router walks on regular trees. When rotating the neighbors in an i.i.d. way the walk is recurrent on the unary tree while it is transient on the d-ary tree, d>2. On the binary tree both cases are possible can be characterized explicitly for periodic sequence. This is deduced from a general criterion for i.i.d. periodic balanced router configurations. The proofs are based on methods from excited random walks connecting the walk's recurrence to a survival problem for a certain branching process.

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

Dr. Sebastian Andres (Universität Bonn)

Title: Quenched invariance principle for the Random Conductance Model in a degenerate dynamic environment

Abstract: In this talk we present a quenched invariance principle for the dynamic random conductance model, that is we consider a continuous time random walk on the integer lattice in an environment of time-dependent random conductances. We assume that the conductances are stationary ergodic with respect to space-time shifts and satisfy some moment condition. One key result in the proof is a maximal inequality for the corrector function, which is obtained by a Moser iteration. This is joint work in progress with A. Chiarini, J.-D. Deuschel and M. Slowik.

**Monday**, 01^{th} February 2016, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Michael Preischl (TUM)

Title: Average percolation

Abstract: We consider d-adic trees with iid edge weights that follow an exponential distribution. Next we fix some value c and look for connected subgraphs that have an average edge weight below c. Now we try to find out whether and for which values of c there are clusters of infinite size. It is proved that there is a phase transition at a certain value c* i.e. for c<c*>c*, we can always find such infinite clusters. Furthermore, we show bounds on the critical value and talk about generalizations. This kind of "average percolation" was first studied by Aldous on complete graphs.</c*>

**Monday**, 08^{th} February 2016, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Carina Geldhauser (Universität Bonn)

Title: The scaling limit of a particle system with long range interaction.

Abstract: We describe the macroscopic behaviourof a coupled bistableparticle system where a large number of particles interact with each other. Due to the properties of the driving force and the noise, the scaling limit does not lead in general to a well-posed equation. We develop conditions on the interaction strength between the particles to ensure existence of solutions to the limiting stochastic PDE. Moreover, we investigate the long-time behaviourof the solution. This is joint work with Anton Bovier.

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