# Colloquium in probability and other talks in summer term 2015

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

**Talks**:

**Monday**, 13^{th} April 2015, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Christoph Temmel (Vrije Universiteit Amsterdam, Netherlands)

Title: Cluster expansion versus disagreement percolation for the hard-sphere model.

Abstract:We consider the hard-sphere model with homogeneous fugacity in d dimensionsal space. We compare two sufficient conditions on the fugacity for uniqueness of the Gibbs measure in the high-temperature regime: a low-fugacity cluster expansion and disagreement percolation. First, we show that there is an upper limit to the cluster expansion, i.e., we derive a necessary condition on the fugacity for a cluster expansion to work. Second, we show that disagreement percolation allows us to derive sufficient conditions for fugacities beyond the cluster expansion regime. Disagreement percolation, introduced by Maes and van den Berg for lattice models, is a method to compare the competing influence of different boundary conditions for finite volume specifications with a product field. This allows to derive the uniqueness of the Gibbs measure from the non-existence of an infinite cluster in site percolation. We adapt disagreement percolation to the point process case to compare with the Boolean model of percolation.

**Monday**, 20

^{th}April 2015, 16:30, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Peter Kevei, Ph.D. (University of Szeged, Hungary)

Title: On the diminishing process of Bálint Tóth.

Abstract: Let K and K_0 be convex bodies in R^d, such that K contains the origin, and define the process (K_n, p_n), n \geq 0, as follows: let p_{n+1} be a uniform random point in K_n, and set K_{n+1} = K_n \cap (p_{n+1} + K). Clearly, (K_n) is a nested sequence of convex bodies which converge to a non-empty limit object, again a convex body in R^d. First, we investigate the process in one dimension, when K = [-1,1]. In this case the limit set is a unit interval, and the distribution of its center has the arcsine law. Further, we study this process for K being a regular simplex, a cube, or a regular convex polygon with an odd number of vertices.

**Monday**, 27^{th} April 2015, 16:30, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Erich Baur (Universität Lyon)

Title: Percolation on large recursive trees.

Abstract: We explain recent results about cluster sizes of Bernoulli bond percolation on large recursive trees. Moreover, we discuss a destruction process on such trees, where edges are cut one after the other in a random uniform order. Partly based on joint work with Jean Bertoin (Zürich).

**Monday**, 4

^{th}May 2015, 16:30, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Benedikt Stufler (LMU)

Title: The continuum random tree is the scaling limit of unlabelled unrooted trees.

Abstract: In the series of seminal papers [Ald91a, Ald91b, Ald93], Aldous showed that the uniform random labelled tree with n vertices and edge lengths rescaled by n

^{-1/2}satisfies a distribu- tional limit called the continuum random tree. He conjectured [Ald91b, p. 55] that conver- gence also holds in the unlabelled setting, i.e. for random Pólya trees and random unlabelled (unrooted) trees. The conjecture for random Pólya trees was confirmed 20 years later by Haas and Miermont [HM12] and a second proof was provided recently by Panagiotou and the speaker [PS15]. The conjecture regarding unlabelled unrooted trees was shown by the speaker in [Stu14] and this talk focuses on giving an overview of the proof and briey providing the necessary background regarding the continuum random tree. The methods involved are based on a combinatorial technique called 'cycle pointing' and its interplay with probability theory via Pólya-Boltzmann samplers. This framework was previously developed by Bodirsky, Fusy, Kang and Vigerske [BFKV11], who applied it in a new proof of the asymptotic enumeration of unlabelled trees.

Monday, 4^{th} May 2015

Graduate Seminar Financial- and Actuarial Mathematics LMU and TUM

**Monday**, 8^{th} June 2015, 16:30, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Prof. Dr. Claudia Klüppelberg (TUM)

Title: Extremes on directed acyclic graphs.

Abstract: We consider a new max-linear structural equation model, where all random variables can be written as a max-linear function of their parents andnoise terms. For the corresponding graph we assume that it is adirected acyclic graph. Leading example are max-stable noise variables,resulting in a max-linear directed acyclic graph. We present basic probabilistic results of our model, in particular characterisations of max-stable DAGs and some properties. We also calculate the regular conditional distributions, from which we can read off the directed Markov property relative to the graph.

**Monday**, 15^{th} June 2015, 16:30, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Diana Conache (Universitaet Bielefeld)

Title: The Uniqueness Problem for Gibbs Fields.

Abstract: We present a generalized and improved version of the Dobrushin-Pechersky criterion for Gibbs fields on graphs. Moreover, by similar techniques, the exponential decay of correlations under the uniqueness conditions can be established.

Monday, 15^{th} June 2015

Graduate Seminar Financial- and Actuarial Mathematics LMU and TUM

**Monday**, 22^{th} June 2015,16:30, LMU, room B 251, Theresienstr. 39, Munich

Stein Bethuelsen (Universiteit Leiden)

Title: Random walks in dynamic random environment

Abstact: Consider a random walk in a random environment which is given by an interacting particle system. In this talk we present new conditions for the random walk to satisfy a law of large numbers and a (quenched) central limit theorem. Our conditions are non-perturbative with respect to the transition kernel of the walker and hold for a large class of dynamics which are uniformly mixing with respect to the starting configuration. Our proof goes by studying "the environment as seen by the walker". By expanding this process backwards in time we obtain existence of an invariant measure absolutely continuous with respect to the underlying process. This is again used to deduce asymptotic properties for the walker. Based on joint work with Florian Völlering (TU Berlin).

**Monday**, 29^{th} June 2015, 16:30, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Christian Döbler (University of Luxembourg)

Title: New error bounds for the random sums CLT by probabilistic methods.

Abstract: Combining Stein's method of normal approximation and probabilistic tools like coupling constructions and the concept of conditional independence, we obtain new abstract results on the distance of the distribution of a random sum of independent random variables and the normal distribution in both the Wasserstein and the Kolmogorov distance. In the case of non-centered summands, our results require a close coupling of the random summation index to its size-biased distribution. We show that for specific distributions of the random index like the Poisson, Binomial or Hypergeoemtric distributions, our general bounds reduce to error bounds of the optimal order.

**Monday**, 6^{th} July 2015,16:30, LMU, room B 251, Theresienstr. 39, Munich

Tobias Gnan (LMU)

Title: TBA

Monday, 13^{th} July 2015

Graduate Seminar Financial- and Actuarial Mathematics LMU and TUM

How to get to Garching-Hochbrück