Oberseminar Stochastik

Abstract:

The dimer model and its associated double-dimer model are fundamental objects in probability theory, statistical mechanics, and combinatorics. While their planar behavior is by now well understood, much less is known beyond planarity.

We study these models on Z^d-like graphs (d≥1), allowing long-range edges whose weights decay with distance. For a large class of such interactions, we show that monomer correlations in the dimer model remain uniformly positive, and that loops in the double-dimer model are macroscopic.

In this talk, I will introduce the models, explain their connection to random permutations, and give an overview of the main proof ideas. In particular, we will take a closer look at the key methods, namely the connection to a spin system and the use of reflection positivity. The project presented in this talk is joint work with Lorenzo Taggi and Wei Wu.

 Einladung

Flavio Dalessi (Uni Basel) 

Branching Random Walk in Random Environment

We introduce the branching random walk in a spatially random branching environment. In this model, particles move according to a continuous-time simple random walk and, independently, branch at spatially dependent random rates. After discussing known properties of the model, we focus on establishing the (quenched) tightness of the maximal displacement around its median. Our approach extends to the discrete-space setting the arguments developed by Černý, Drewitz, and Oswald (2025) to prove tightness of the maximum of the branching Brownian motion in random environment. Specifically, we exploit the connection between the branching random walk and the randomized discrete Fisher-KPP equation through a study of the latter under different initial conditions.

Einladung

Janine Piesold (JGU)

On a Branching Annihilating Random Walk

We consider a discrete-time branching annihilating random walk (BARW). Within a time step, each particle, before dying, produces a random number of offspring which are then randomly and independently displaced in space. If, after the displacement, a site is occupied by several particles, all particles at that site are annihilated. This can be thought of as a very strong form of local competition and entails that the system is not monotone.
The motivation of our current discussion stems from the long-term behaviour of this model on Z^d. For a Poissonian number of offspring and a uniform displacement within a ball of finite radius around the parent’s position, survival and complete convergence results were shown by Birkner et al. (2024). While a survival result was obtained for a more general class of offspring distributions, the extension of the complete convergence result to general offspring laws has proven to be more intricate. To gain more insight on this process, we further investigate the mean-field version of the BARW with general offspring law. In this talk, we will thus explore the BARW on the complete graph and highlight its connection to a deterministic dynamical system as well as a classical urn occupancy problem. We then discuss a local central limit result on the related urn occupancy problem based on Stein’s method. This result enables us to construct couplings, which potentially lay the foundation to prove the complete convergence result for the spatial model.

Lars Schroeder (Universität Twente) 

Stationary distribution of node2vec random walks

node2vec random walks are tuneable random walks that come from the popular algorithm node2vec which is used for network embedding. The transition probabilities of the random walks depend on the previous visited node and on the triangles that contain the current and the previous node. Even though the algorithm is widely used in practice, mathematical properties of node2vec random walks almost have not been investigated. We present results on the stationary distribution for household models (graphs with clique-structured communities) by studying a coupling to a random walk that jumps in between communities and results on regular graphs by going to a higher-order space. Joint work with Clara Stegehuis, Gianmarco Bet and Luca Avena.

Noela Müller  (Universität Eindhoven) - ENTFÄLLT LEIDER KURZFRISTIG !!!

Sharp thresholds and hitting times for factors in the Binomial random graph

Louis Chataignier (Université Toulouse III)

Upper moderate deviation probabilities for the maximum of a branching random walk

In 2013, Elie Aı̈dékon obtained the convergence in distribution, as time n goes to infinity, of the maximum of a supercritical branching random walk, once recentered by an explicit function m(n). More recently, Lianghui Luo gave an asymptotic equivalent for the upper large deviation probabilities of this maximum. In this talk, I will present a joint work with Lianghui Luo in which we study an intermediate regime. We obtain an asymptotic equivalent for the probability that the maximum arrives at distance x(n) above m(n), where 1 << x(n) = O(sqrt(n)).

Einladung

Sommersemester 2024

Prof. Dr. Steffen Dereich (WWU Münster)

Einladung und Abstract

Stein Andreas Bethuelsen (University of Bergen)

Einladung und Abstract

Samuel Modee (University of Bergen)

Einladung und Abstract

Timo Schlüter (Universität Basel)

Einladung und Abstract

Wintersemester 2023/24

Dr. Marco Seiler, Frankfurt

Einladung und Abstract