# Oberseminar Wahrscheinlichkeitstheorie und andere Vorträge im Wintersemester 2019/20

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

**Talks**:

**Monday**, 21^{th} October 2019, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Amitai Linker (Universidad de Chile)

Title: The contact process on stationary dynamic networks

Abstract: The contact process is a particle system introduced by Ted Harris as a simple toy model for the spread of disease on a network. Governed by simple rules, this process has been widely studied over more than 40 years on a number of different networks such as lattices, trees, the configuration model and the preferential attachment model, to name a few. We will address the question of survival and extinction on each of these examples, and then ask what could be the effect of adding some stationary dynamics to the networks. We will show two examples in which the addition of dynamics hurt and benefit the infection, respectively. (Joint work with E. Jacob, P. Mörters and D. Remenik)

**Monday**, 28^{th} October 2019, 16:30, TUM, room 2.02.01, Parkring 11, Garching-Hochbrück (Technische Universität München)

Anders Öberg (Uppsala University)

Title: Uniqueness of g-measures and existence of eigenfunctions for the Dyson model

Abstract: There are several conditions for uniqueness of g-measures (stationary distributions for chains with dependence on the infinite past) which will reviewed during the talk. These conditions suggests that there is a problem of finding an eigenfunction of the transfer operator for general potentials even in the simplest examples of one-dimensional statistical physics, if not quite strong regularity conditions of the potentials are imposed, such as that of summable variations. This is also reflected in the problem of studying the the relation between the equilibrium measures of the one-sided and two-sided long-range models. For the Dyson model we can prove that there exists an eigenfunction if the inverse critical temperature is strictly smaller than the critical value. More generally, we have reasons to believe that a certain condition by Berbee is the appropriate condition

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

Stephan Mertens ( OVGU Magdeburg and Santa Fe Institute )

Title: Percolation: New results for an old problem

Abstract: It is known that the percolation threshold for percolation on Z^d can be written as an asymptotic series in 1/d, for both bond and site percolation. The first four terms of both series were computed in the 1970s, but the series have not been extended since then. I will show how brute-force enumerations, combinatorial identities and a new approach based on Padé approximants can be deployed to compute more terms of the series.In the second part of the talk I will present a startling symmetry in the number of percolating configurations. This symmetry implies, for example, that the number of percolating configurations on finite subsets of Z^d is always odd.

**Monday**, 11^{th} November 2019, 16:30, LMU, room B252, Theresienstr. 39, Munich

Dominik Fickenwirth (LMU “MSc presentation”)

Title: Cluster Expansion

Abstract: We definite a general measure theoretical setting for the cluster expansion method and study a criterion for its convergence.

**Monday**, 25^{th} November 2019, 16:00, LMU, room B251, Theresienstr. 39, Munich

Noela Müller (Goethe-Universität Frankfurt)

Title: The replica symmetric phase of random constraint satisfaction problems

Abstract: Random constraint satisfaction problems play an important role in computer science and combinatorics. For example, they provide challenging benchmark instances for algorithms and they have been harnessed in probabilistic constructions of combinatorial structures with peculiar features. In an important contribution, physicists made several predictions on the precise location and nature of phase transitions in random constraint satisfaction problems. Specifically, they predicted that their satisfiability thresholds are quite generally preceded by several other thresholds that have a substantial impact both combinatorially and computationally. These include the condensation phase transition, where long-range correlations between variables emerge, and the reconstruction threshold. In this paper we prove these physics predictions for a broad class of random constraint satisfaction problems. Additionally, we obtain contiguity results that have implications on Bayesian inference tasks, a subject that has received a great deal of interest recently. This is joint work with Amin Coja-Oghlan and Tobias Kapetanopoulos.

**Im Anschluss daran:**

**Monday**, 25^{th} November 2019, 17:30, LMU, room B251, Theresienstr. 39, Munich

Wolfgang König (WIAS und TU Berlin)

Title: A large-deviations approach to the multiplicative coalescent

Abstract: We consider the (non-spatial) coalescent model (sometimes called the Marcus-Lushnikov model), starting with $N$ particles with mass one each, where each two particles coalesce after independent exponentially distributed times. The corresponding coagulation kernel ist multiplicative in the two masses, hence the coalescent is also called multiplicative. There are strong relations with the time-dependent Erd\H{o}s-R\'enyi graph. We work in the thermodynamic limit $N\to\infty$ at a fixed time $t$ and derive a joint large-deviations principle for all relevant quantities (microscopic, mesoscopic and macroscopic particle sizes) with an explicit rate function. We deduce laws of large numbers and in particular derive from that the well-known phase transition at time $t=1$, the time at which a macroscopic particle appears, as well as the well-known Smoluchowski characterisation of the statistics of the finite-sized particles. (joint work with Luisa Andreis and Robert Patterson.)

**Monday**, 2^{th} December 2019, 16:30, LMU, room B252, Theresienstr. 39, Munich

Maria Infusino (Universität Konstanz)

Title: Projective limit techniques for the infinite dimensional moment problem

Abstract: In this talk we give an introduction to the infinite dimensional moment problem,

i.e. the problem of establishing if a measure supported in an infinite dimensional

space (e.g. a function space) can be described from its moments. Although infinite

dimensional moment problems have a long history, their theory is not as well

developed as in the finite dimensional case and is still not up to the demand of the

applications. We will therefore focus on the following general moment problem: when

can a linear functional on an infinitely generated algebra A be represented as an

integral w.r.t. a Radon measure on the space X(A) of all homomorphisms on A equipped

with the Borel sigma-algebra generated by the weak topology? Our main idea is to

construct X(A) as a projective limit of all X(S) with S finitely generated

subalgebra of A, so to be able to exploit the results for the classical finite

dimensional moment problem in the infinite dimensional case. In fact, we show that

under a Prokhorov-like assumption, the infinite dimensional moment problem on A is

solvable iff for any finitely generated subalgebra S of A the corresponding finite

dimensional moment problem is solvable. We will provide some applications of this

result to the case when A is the algebra of polynomials in infinitely many variables

or the tensor algebra of an infinite dimensional vector space, showing the power of

this approach as well as its potential in attacking the realizability problem in

statistical mechanics. (Joint work with Salma Kuhlmann, Tobias Kuna and Patrick

Michalski)

**Monday**, 9^{th} December 2019, 16:30, LMU, room B252, Theresienstr. 39, Munich

Frank Proske (Universität Oslo)

Title: Strong solutions of SDE's with generalized drift and multidimensional fractional Brownian initial noise

Abstract: In this presentation we discuss a new method for the construction of strong solutions to a SDE with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters H<1/2. Here the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion.

Our approach for obtaining strong solutions relies on techniques from Malliavin calculus combined with a "local time variational calculus" argument.

**Thursday**, 12^{th} December 2019, 16:30, TUM, room B045, Theresienstr. 39, Munich

Christina Zou (Universität Oxford)

Title: A free boundary representation of Root's solution to the Skorokhod embedding problem for Markov processes

Abstract: A classical problem in stochastic analysis is the Skorokhod embedding problem: Given a Brownian motion and a probability measure, the task is to stop the trajectories of the process such that the terminal points are distributed according to the given measure. One approach in order to determine a solution for the problem is to construct it as a first hitting time of the Root barrier. Rost proved in 1976 that Root’s solution has the minimal variance among the solutions to Skorokhod embedding problem using methods from probabilistic potential theory. We are going to investigate sufficient conditions such that such an embedding can be made and provide a free boundary characterisation of Root’s solution for a general class of Markov processes.

**Monday**, 16^{th} December 2019, 16:30, TUM, room 2.02.01, Parkring 11, Garching-Hochbrück (Technische Universität München)

Florian Theil (Warwick)

Title: Effective Thermal Electrochemical Model for Porous Electrode Batteries

Abstract: Thermal electrochemical models for porous electrode batteries (such as

lithium ion batteries) are widely used. Due to the multiple scales involved, solving the model

accounting for the porous microstructure is computationally expensive, therefore effective

models at the macroscale are preferable. However, these effective models are usually postulated

ad hoc rather than systematically upscaled from the microscale equations. Using the method

of periodic homogenisation I will demonstrate that in a suitable limit the equations converge to an

effective thermal electrochemical model as the fineness of the microstructure converges to 0.

**Monday**, 13^{th} January 2020, 16:30, LMU, room B252, Theresienstr. 39, Munich

Stefan Wagner (Universität Ulm)

Title: Bakry-Émery theory and the kinetic Fokker-Planck equation

Abstract: In this talk we will explore the gamma calculus for Markov semigroups. We will introduce the Bakry-Émery curvature condition and deduce gradient bounds, Poincaré inequalities and log-Sobolev inequalities.

The goal is to show exponential fast convergence to an equilibrium. We want to apply the same in the theory of PDEs. It turns out that the curvature condition is not fulfilled by the kinetic Fokker-Planck equality.

Therefore we introduce a new calculus and a new curvature condition.

**Thursday**, 16^{th} January 2020, 16:30, LMU, room A027 , Theresienstr. 39, Munich

Dinh Thanh Nguyen (LMU Master presentation)

Title: Sharp Phase Transition in the Random Connection Model

Abstract: This work explores some properties of the random

connection model. Using modern results in continuum percolation, we

give a proof of the equality of the percolation critical value and

the susceptibility critical value by adapting a method of Aizenman

and Barsky from 1987. This demonstrates a result of Meester in 1987

anew and in a more straightforward way. Further, we prove the sharp

phase transition for the random connection model under the

assumption that its connection function has finite support. Here, we

utilize a method developed by Duminil-Copin, Raoufi and Tassion.

Finally, we expose and discuss some difficulties when applying the

method to random connection models with other connection functions,

for instance, exponentially decaying ones.

**Monday**, 27^{th} January 2020, 16:30, LMU, room B252, Theresienstr. 39, Munich

Dimitrios Tsagkarogiannis (University of L'Aquila)

Title: Nonequilibrium fluctuations for current reservoirs

Abstract: Stationary non equilibrium states are characterized by the presence of steady currents flowing through the system as a response to external forces. We model this process considering the simple exclusion process in one space dimension with appropriate boundary mechanisms which create particles on the one side and kill particles on the other. The system is designed to model Fick's law which relates the current to the density gradient. In this talk we focus on the fluctuations around the hydrodynamic limit of the system. The main technical difficulty lies on controlling the correlations induced by the boundary action. This is work in progress jointly with Panagiota Birmpa and Patricia Gonçalves.

How to get to Garching-Hochbrück