# Colloquium in probability and other talks in summer term 2018

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

**Talks**:

**Monday**, 9^{th} April 2018, 15:00, LMU, room B252, Theresienstr. 39, Munich

Dr. Tal Orenshtein (Humboldt Universität Berlin)

Title:Critical wetting models in 1+1 dimensions

Abstract: Wetting models are a class of polymer models enjoying an interplay between two forces - local (pinning) and global (presence of hard wall). In 1+1 dimensions, the standard case, AKA delta-pinning model, is completely solved and a sharp phase transition holds. Indeed, the asymptotic behavior of the system is drastically different in the sub-critical, the super critical and the critical phases. In particular, at criticality, the rescaled path converges in law to the reflected Brownian motion. However, the pinning potential has strong a singularity at zero. Considering the model on a strip of fixed size and with a constant pinning function, the existence of phase transition and off-critical scaling limits are known. In the talk we shall discuss a recent work with Jean-Dominique Deuschel attempting to tackle the singularity while keeping the same critical behavior. In particular, we will present our result on a path scaling limit for the strip model so that the strip is shrinking to zero, where any smooth pinning function is allowed as long as it is an approximation of the critical value of the standard model.

**Monday**, 9^{th} April 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Roland Bauerschmidt (University of Cambridge)

Title: Renormalisation of classical spin systems

Abstract: I will give an overview of recent results on classical spin systems. Spin systems are prototypical models for phase transitions. Their complex behaviour has turned out to be partially accessible from a large variety of complementary points of view, and has made them objects are intense mathematical study. I will provide an impression of these, with focus on rigorous renormalisation as a powerful method. The talk is based on joint works with David Brydges and Gordon Slade, with Gordon Slade and Ben Wallace, and with Thierry Bodineau.

**Monday**, 16^{th} April 2018, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Piotr Dyszewski (Univeristy of Wroclaw)

Title: Large deviations for a random walk in random environment revisited

Abstract: We will consider a random walk on integers in site-dependent random environment. We will show how to use the associated branching process to obtain a precise large deviation asymptotic for the random walk. The talk is based on a joint work with Dariusz Buraczewski.

**Monday**, 23^{th} April 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Alexander Schiefner (LMU)

Title: Degree distribution of conditioned Galton-Watson trees

Abstract: In this presentation, we want to examine the probability that a Galton-Watson tree with n nodes, has exactly d nodes with k children and find a local limit theorem for it. Therefore, I will give a short introduction about the conditioned Galton-Watson trees, summaries some of their properties and explain, which restrictions we make for this examination. Furthermore, I will present some already known approximations, before I state and roughly proof the new form of the local limit theorem.

**Monday**, 7^{th} May 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Sascha Kissel (Ruhr-Universität Bochum)

Title: Discrete Hard and Soft-Core Widom-Rowlinson model under stochastic time evolution

Abstract: In this talk, we consider the discrete Widom-Rowlinson model with Hard-Core constraint and the related Soft-Core model. First we proof Dobrushin Uniqueness for the Soft-Core model and with this result we proof that the time evolved measure under stochastic symmetric spin flip is a Gibbs measure for small times. In the second part we talk about the immediate loss and recovery of the Gibbs property of the time evolved Hard-Core Widom-Rowlinson model.

**Monday**, 28^{th} May 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Simon Stadler (LMU)

Title: On propagation of chaos for a stochastic, proliferating particle system

Abstract: In this talk, we want to give an overview of our proof for the mean field limit and propagation of chaos regarding the following particle system. Consider N particles in dimension 3 acting via a Brownian motion, proliferation and a pair interaction force scaling like $ \frac{1}{|x|^{\lambda}}, \lambda < 2 $ with cut-off width $N^{- \frac{1}{3}} $. Proliferation times of the particles are exponentially distributed which leads to Poisson processes for the number of proliferation events. The proof we present is based on a Gronwall argument to control the distance between the (exact) microscopic dynamics and the approximate mean field dynamics. **Im Anschluss daran** Simon Bischoff (LMU)

Title: A Generalization of Mean Field Limits for Dynamical Systems

Abstract: The talk is about a comparison of many particle systems. While in one system the particles move independently, in the other an interaction force acts on those. Solving the vlasov equation, under certain assumptions we will show that those two systems behave similiar in a suitable sense. Apart from that we will have a look at some assumptions concerning the interaction force to get the result of similiar behavior.

**Monday**, 4^{th} June 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Stein Andreas Bethuelsen (TU Munich)

Title: Loss of memory and continuity properties for the supercritical contact process

Abstract: The contact process is an interacting particle system which models the spread of an infection in a population. In this talk I will focus on the evolution of this process in the supercritcal regime within a partial (and finite) subspace of the population. In particular, I will present some recent results on the loss of memory property for such partially observed processes and discuss their continuity properties with respect to conditioning (in the sense of g-measures). Several motivations for studying such projections, and some open questions, will be discussed during the talk.

**Thursday**, 7^{th} June 2018, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Prof. Dr. Arnaud Le Ny (Universite Paris-Est)

Title: Interface states for long-range Ising models

Abstract: Interface states, also called Dobrushin states, are extremal non-translation invariant infinite volume Gibbs measures considered to model rigid interfaces, and originally doscovered by Dobrushin in 1972 in the context of nearest-neighbours Ising model in absence of external fields. In this talk, we shall review old and more recent results for both nearest-neighbours and long-range polynomially decaying pair potentials and investigate the rigidity of the interface in the particular two-dimensional case, in both isotropic and anisotropic long-range cases. This is a joint work with L. Coquille (Grenoble), A. van Enter (Gronigen) and W. Ruszel (Delft).

**Monday**, 18^{th} June 2018, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Prof. Dr. Claudia Klüppelberg (TUM)

Title: Max-linear models on infinite graphs generated by Bernoulli bond percolation

Abstract: We extend finite-dimensional max-linear models to models on infinite graphs, and investigate their relations to classical percolation theory, more precisely to nearest neighbour bond percolation. We focus on the square lattice $\mathbb{Z}^2$ with edges to the nearest neighbours, where we direct all edges in a natural way (north-east) resulting in a directed acyclic graph (DAG) on $\mathbb{Z}^2$. On this infinite DAG a random sub-DAG may be constructed by choosing vertices and edges between them at random. In a Bernoulli bond percolation DAG edges are independently declared open with probability $p\in [0,1]$ and closed otherwise. The random DAG consists then of the vertices and the open directed edges. We find for the subcritical case where $p\le 1/2$ that two random variables of the max-linear model become independent with probability 1, whenever their distance tends to infinity. In contrast, for the supercritical case where $p>1/2$ two random variables are dependent with positive probability, even when their node distance tends to infinity. We also consider changes in the dependence properties of random variables on a sub-DAG $H$ of a finite or infinite graph in $\mathbb{Z}^2$, when enlarging this subgraph. The method of enlargement consists of adding nodes and edges of Bernoulli percolation clusters. Here we start with $X_i$ and $X_j$ independent in $H$, and answer the question, whether they can become dependent in the enlarged graph. As a possible application we discuss extreme opinions in social networks The talk is based on joint work with Ercan Sönmez and the following paper. [1] Klüppelberg, C. and Sönmez, E. (2018) Max-linear models on infinite graphs generated by Bernoulli bond percolation. In preparation.

**Monday**, 25^{th} June 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Christian Mönch (Universität Mannheim)

Title: Persistence of self-similar process with stationary increments

Abstract: The persistence probability $P(T)$ of a real valued process is the probability that the process stays below a given value up to time $T$. It is conjectured that $P(T)=T^{-(1-H)+o(1)}$ for any $H$-self-similar process with stationary increments and continuous paths. Until recently, this had only been rigourously verified for Fractional Brownian Motion (Molchan, Commun. Math. Phys., 1999). I will discuss the case where the processes under consideration are Hermite processes -- close relatives of fractional Brownian motion, albeit in general non-Gaussian. As a tool, I will present a decorrelation inequality, which is reminiscent of Slepian's lemma for Gaussian processes and may be of independent interest. The talk is in large parts based on the article {https://arxiv.org/abs/1607.04045} with Frank Aurzada.

**Monday**, 2^{th} July 2018, 16:30, LMU, room B252, Theresienstr. 39, Munich

Elias Haslauer (LMU)

Title: Uncertainty Models and Ergodicity

Abstract: We study two approaches to model uncertainty ? capacities and nonlinear respectively sublinear expectations. Originally arisen in different contexts, both theories have developed widely independently and barely refer to each other. We show that both approaches are in fact the same, more precisely, there is an equivalence of capacities and nonlinear expectations as well as of convex continuous capacities and regular sublinear expectations. These findings can be fruitful for both theories and suggest the development of a unified theory. In addition, we discuss two recently established ergodic theorems for capacities respectively sublinear expectations. In light of the established equivalences a slight improvement of both statements is possible. **Im Anschluss daran** Harald Koppen (LMU, MSc presentation)

Title: Sharpness of the Phase Transition in the Contact Process

Abstract: The contact process is an interacting particle system which can be interpreted as the spread of an infection. In this talk, we focus on the contact process on the two-dimensional integer lattice and consider the percolation transition. That is, we examine the size of the occupied cluster of the origin subject to the upper invariant measure. It is well known that there is a critical value \lambda_c such that for all infection rates bigger than \lambda_c, the upper invariant measure is non-trivial. Furthermore, there is another critical value \lambda_p such that the probability of the aforementioned cluster being infinite is bigger than zero for all infection rates bigger than \lambda_p. However, if the infection rate is smaller than \lambda_p, the distribution of the size of the cluster has an exponential tail. We sketch a proof of this result using techniques as in the proof of a related result for confetti percolation. A short introduction to the contact process will be given at the beginning of the talk.

**Monday**, 9^{th} July 2018, 16:45, LMU, room B252, Theresienstr. 39, Munich

Franziska Flegel (Weierstraß-Institut Berlin)

Title: Spectral localization vs. homogenization in the random conductance model

Abstract: We study the asymptotic behavior of the top eigenvectors and eigenvalues of the random conductance Laplacian in a large domain of Z^{d} (d≥2) with zero Dirichlet conditions. Let the conductances w be positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. Then we show that the spectrum of the Laplacian displays a sharp transition between a completely localized and a completely homogenized phase. A simple moment condition distinguishes between the two phases.

In the homogenized phase we can even generalize our results to stationary and ergodic conductances with additional jumps of arbitrary length. Here, our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence. The investigation of the homogenized phase is joint work with M. Slowik and M. Heida.**Im Anschluss daran, 17:45 Uhr **(in Kooperation mit dem Oberseminar “Dynamics”),

Prof. Dr. Sabine Jansen (LMU)

Title: Metastability and effective interfaces for the high-intensity Widom-Rowlinson model

Abstract: Consider a diffusion $X_t$ in an energy landscape $U(x)$, i.e., a solution to the stochastic differential equation $d X_t = - \nabla U(X_t) d t + \sqrt{\eps} d B_t$. In the small-noise limit $\eps \searrow 0$, the diffusion started in a local energy minimum exhibits metastable behavior - it takes a long time to reach the global minimum. The answer to the question "how long" is provided by the Eyring-Kramers law. The talk addresses similar questions for a Markov birth and death process of points in $\mathbb R^d$, where the energy landscape is replaced with the rate function of some suitable large deviations principle and the analogue of the Eyring-Kramers law brings in functional central limit theorems and infinite-dimensional Gaussians. Based on joint work in progress with Frank den Hollander, Roman Kotecky, and Elena Pulvirenti.

**Thursday**, 5^{th} Juli 2018, 14:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Johannes Bäumler (TUM)

Title: Uniqueness and non uniqueness for spin-glass ground states on trees

Abstract: We consider a Spin Glass at temperature T = 0 with gaussian couplings, where the underlying graph is a locally finite tree. We prove that uniqueness of ground state pairs is equivalent to recurrence of the simple random walk on the tree. Furthermore we give a sufficient condition for the above statements.

**Tuesday**, 10^{th} July 2018, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Mykhaylo Shkolnikov (Princeton University)

Title: Particles interacting through their hitting times: neuron firing, supercooling and systemic risk

Abstract: I will discuss a class of particle systems that serve as models for supercooling in physics, neuron firing in neuroscience and systemic risk in finance. The interaction between the particles falls into the mean field framework pioneered by McKean and Vlasov in the late 1960s, but many new phenomena arise due to the singularity of the interaction. The most striking of them is the loss of regularity of the particle density caused by the self-excitation of the system. In particular, while initially the evolution of the system can be captured by a suitable Stefan problem, the following irregular behavior necessitates a more robust probabilistic approach. Based on joint work with Sergey Nadtochiy.

How to get to Garching-Hochbrück