# Oberseminar Wahrscheinlichkeitstheorie und andere Vorträge im SS 2011

**Veranstalter:**Nina Gantert (TUM), Hans-Otto Georgii (LMU), Franz Merkl (LMU), Thomas Richthammer (LMU), Silke Rolles (TUM), Vitali Wachtel (LMU) Gerhard Winkler (Helmholtz Zentrum München)

- Montag, 9.5.2011, 16:15, Raum B251 an der LMU.

Simon Aumann (LMU):*Zählmaß pivotaler Punkte*

**Abstract:**Es wird der Artikel "Pivotal, cluster and interface measures for critical planar percolation" von Christophe Garban, Gabor Pete und Oded Schramm, arXiv:1008.1378, 2010, vorgestellt. Sie definieren ein Maß, das pivotale Punkte kritischer Perkolation zählt und zeigen, dass es im Limes Gitterweite gegen Null gegen ein Maß konvergiert, das aus der kritischen Perkolation konstruiert werden kann. Dies ist ein wesentlicher Schritt für den Beweis, dass nahkritische Skalenlimiten existieren. - Montag, 16.5.2011, 16:15, Raum MI 03.10.011 an der TUM

Mihyun Kang:*Phase transitions in random graphs*

**Abstract:**The phase transition deals with sudden global changes and is observed in many fundamental problems from statistical physics, mathematics and theoretical computer science, for example, Potts models, graph colourings and random k-SAT. The phase transition in random graphs refers to a phenomenon that there is a critical edge density, to which adding a small amount a drastic change in the size of the largest component occurs. In Erdös-Renyi random graph, which begins with an empty graph on n vertices and edges are added randomly one at a time to a graph, a phase transition takes place when the number of edges reaches n/2 and a giant component emerges. Since this seminal work of Erdös and Renyi, various random graph models have been introduced and studied. In this talk we discuss phase transitions in several random graph models, including Erdös-Renyi random graph, random graphs with a given degree sequence, random graph processes and random planar graphs. - Montag, 6.6.2011, 16:15, Raum MI 03.10.011 an der TUM

Sebastian Müller^{}(Marseille):*Branching random walks on groups*

**Abstract:**The theory of branching random walks (BRW) on groups is still maiden-like. We survey basic and recent results on BRW and propose some accessible open problems.

A BRW is a system of particles evolving as follows. The process starts with one particle in the group origin. Then at each (discrete) time step a particle branches according to some offspring distribution (with mean m) and moves one step according to an underlying random walk. A BRW is called recurrent if the origin is visited by infinitely many particles with positive probability and transient otherwise. As a consequence of Kesten's amenability criterion any BRW with m>1 is recurrent. There is a phase transition for BRW on non-amenable groups, i.e., there exists some m_c>1 such a BRW with m>m_c is recurrent and with mThe talk does not require any specific knowledge on group theory nor on probability theory and is designed for a wider audience. - Mittwoch, 29.6.2011 um 17:15 in Raum PR 2.01.11 in Garching-Hochbrück

Peter Mörters^{}:*The giant component in preferential attachment networks*

**Abstract:**We study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function of its current degree. We use approximation by branching random walks to find necessary and sufficient criteria for the existence and robustness of a giant component in these networks.

The talk is based on joint work with Steffen Dereich (Marburg). - 10. Erlanger-Münchner Tag der Stochastik on Friday, July 1, 2011
- Montag, 11.7.2011, 16:15, Raum MI 03.10.011 an der TUM

Janos Engländer^{}(University of Colorado):*Some challenging open problems for spatial branching models*

**Abstract:**I will review some spatial branching models in random environments and with interactions and suggest some (hopefully) interesting open problems. The random environment model is joint work with N. Sieben who provided the simulations. - Workshop Women in probability July 15-16, 2011
- Montag, 25.7.2011, 16:15, Raum MI 03.10.011 an der TUM.

Noam Berger^{}(Berlin and Jerusalem):*Lack of percolation for low temperature spin glasses in two dimensions*

**Abstract:**In the talk I will define the Edwards-Anderson Spin glassmodel. I will then show that in two dimensions and in all low enough temperatures,for translation covariant Gibbs measures, almost surely the unsatisfied edges do not percolate.This is joint work with Ran Tessler. - Montag, 25.7.2011, 17:15, Raum MI 03.10.011 an der TUM.

Andrea Collevecchio:*On a preferential attachment and generalized Polya's urn model*

**Abstract:**We study a general preferential attachment and Polya's urn model. At each step a new vertex is introduced, which can be connected to at most one existing vertex. If it is disconnected, it becomes a pioneer vertex. Given that it is not disconnected, it joins an existing pioneer vertex with a probability proportional to a function of the degree of that vertex. This function is allowed to be vertex dependent, and is called reinforcement function. We prove that there can be at most three phases in this model, depending on the behavior of the reinforcement function. Consider the set whose elements are the vertices whose cardinality tends a.s. to infinity. We prove that this set either is empty, or it has exactly one element, or it contains all the pioneer vertices. Moreover, we describe the phase transition in the case where the reinforcement function is the same for all vertices. Our results are general, and in particular we are not assuming monotonicity of the reinforcement function. Finally, consider the regime where exactly one vertex has a degree diverging to infinity, and suppose that at a certain stage, a given vertex has a large degree compared to the others. We give a lower bound for the probability that this vertex will be the leading one, i.e. its degree will diverge to infinity. Our proofs rely on a generalization of the Rubin construction given for edge-reinforced random walks, and on a Brownian Motion embedding.

This is joint work with Codina Cotar and Marco Li Calzi.

How to get to Garching-Hochbrück