The goal of this conference is to bring together young and senior researchers working in the field of branching processes, with a special focus on branching random walks and branching Brownian motion. We aim to create a stimulating atmosphere for the exchange of ideas and to foster new collaborations. There will be plenty of time for discussions.
building S2 07, Hochschulstraße 6, 64289 Darmstadt
ABSTRACT
Stochastic geometry is a classical topic in probability theory and has seen interesting developments in past years. The aim of this school is to introduce PhD students and young post-docs to some aspects of this development. The core of the school are two mini courses, each consisting of five 90-minute lectures:
Zakhar Kabluchko (Münster) (Abstract) Joseph Yukich (Lehigh) (Abstract+Lecture notes).
The mathematical study of Complex Networks is an active subject in probability with many real world applications. The aim of this school is to introduce PhD students and young post-docs to some recent developments in this field. The core of the school consists of two mini courses, each consisting of five 90-minute lectures by
Remco van der Hofstad (Eindhoven) Frank den Hollander (Leiden).
Die Modelle zufälliger räumlicher Permutationen und Schlaufen lassen sich gut anschaulich erklären, und enthalten leicht zu formulierende, aber sehr schwierig zu lösende Probleme.
Als einfachstes Beispiel sei U eine (große) endliche Teilmenge des Gitters Z^d. Zwischen den Punkten von U fügt man nun Kanten so ein, dass jeder Punkt entweder isoliert (keine Kante) ist, oder genau zwei Kanten an ihn angrenzen. Weiter darf man Kanten nur zwischen Punkten einfügen, die im Gitter U benachbart sind. Offenbar ergibt sich hierdurch eine Sammlung von isolierten Punkten und geschlossenen “Schlaufen”. Unter allen (endlich vielen) Möglichkeiten, eine durch obige Regel erlaubte Konfiguration von Kanten zu erzeugen, wählt man nun zufällig eine aus, allerdings nicht gleichverteilt, sondern mit einer Zähldichte, die proportional zu $\exp(- \alpha K)$ ist, wobei K die Anzahl der eingezogenen Kanten ist. Eine (ungelöste) Frage ist beispielsweise, wie sich (im Grenzwert unendlich großer Gebiete U) die erwartete Länge der Schlaufe verhält, die den Punkt $0 \in \bbZ^d$ enthält.
In meinem Vortrag werde ich verschiedene Varianten dieser Modelle vorstellen und auf einige ihrer vermuteten und bewiesenen Eigenschaften eingehen. Ebenso werde ich kurz über die auf den ersten Blick überraschenden Beziehungen zur Theorie der Bose-Einstein-Kondensation sprechen. Der Vortrag richtet sich an ein allgemeines mathematisches Publikum und kann ohne weitreichende Kenntnisse der Wahrscheinlichkeitstheorie gut verstanden werden.
The talk will be given in English if there is demand.
Institut für Mathematik UIBK, Technikerstraße 13, 7. OG - Seminarraum 734
ABSTRACT
Brownian motion is one of the pillars of statistical physics with applications ranging from astrophysics to biological physics. The theoretical foundation is well understood since Einstein and Smoluchowski introduced a probabilistic interpretation to derive diffusion as a macroscopic law. In modern language, the diffusion propagator follows from the central limit theorem. In stochastics the paradigm of Brownian motion sets the foundation for virtually all stochastic processes in continuous time.
Interestingly, in the experimental realization of a suspended mesosized particle in a simple fluid, which triggered the theoretical development, the underlying dynamics does not fulfill completely the assumptions of Brownian motion. This is trivially obvious at the shortest time scales, where the molecular nature of the surrounding fluid becomes relevant. However, quite surprisingly, even at long times, the dynamics displays hidden long-range correlations which only algebraically fade out.
In the presentation I will provide 'physicist's view' on Brownian motion including some historic remarks on the key ideas of the modern theoretical description. Recent experimental progress [1,2] has allowed to direclty monitor the subleading correlations in the mean-square displacement which have been anticipated long time ago by hydrodynamic arguments.
[1] T. Franosch et al., Nature 478, 85-88 (2011)
[2] S. Jeney et al., Phys. Rev. Lett. 100, 240604 (2008)
We study the amount of information that is contained in "random pictures“, by which we mean the sample sets of a Boolean model. The talk will explain the concept of a Boolean model, which includes a model for a "completely random distribution of points in the plane". Then we shall study different concepts making the term "amount of information“ mathematically precise.
The presented results are based on a joint work with Mikhail Lifshits (St. Petersburg).
TITLE
Branching random walks and the Fisher-KPP equation
Branching random walks and branching Brownian motion are stochastic processes which have been receiving a lot of attention during the last couple of decades. They appear in a variety of more applied sciences (such as biology, statistical mechanics, and computer science) and exhibit deep connections to partial differential equations.
We will provide a survey of some classical results and discuss the interesting effects of extensions to spatially heterogeneous branching rates.
TITLE
Large deviations and distances in the spatial preferential attachment model
Preferential attachment models are self-reinforced growth models for complex networks. Recently, there has been much interest in developing PA models for networks embeded into some underlying geometric space. For one such model, proposed by Jacob and Mörters, I will discuss a Large Deviation principle for empirical vertex neighbourhoods and the asymptotics of the distance of two typical vertices in the giant component. The talk is based on joint work with Christian Hirsch (Aalborg).
