# Oberseminar Wahrscheinlichkeitstheorie und andere Vorträge im Wintersemester 2021/22

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

**Talks**:

**Monday**, 11^{th} October 2021, 16:30, LMU, room B252, Theresienstr. 39, Munich

Hybrid meeting: Synchronous broadcast via zoom via the link (Click here for Zoom Meeting)

Detlev Kreß (LMU; MSc presentation)

Title:** **A percolation model without positiv correlattion

Abstract: We introduce a bond-percolation model that is a modification of the corrupted compass model introduced by Christian Hirsch, Mark Holmes and Victor Kleptsyn (2021).On a given graph we start in each vertex independent with probability p a random walk of length L. We make an edge occupied if it was used by a random walk. This model does not exhibit positive correlation.If L is choosen such that there is percolation for p=1, we have a sharp phase transition for p. We discuss the question of percolation on the hypercubic lattice and show that on the square lattice percolation occurs for L=2.

**Monday**, 25^{th} October 2021, 16:30, LMU, room B252, Theresienstr. 39, Munich

Hybrid meeting: Synchronous broadcast via zoom via the link (Click here for Zoom Meeting)

Wolfgang Löhr (Universität Duisburg-Essen)

Title:** **A new state space of algebraic measure trees for stochastic processes

Abstract: In the talk, I present a new topological space of ``continuum'' trees,

which extends the set of finite graph-theoretic trees to uncountable

structures, which can be seen as limits of finite trees. Unlike previous

approaches, we do not use the graph-metric but formalize the tree-structure

by a tertiary operation on the tree, namely the branch-point map. The

resulting space of algebraic measure trees has coarser equivalence classes

than the more classical space of metric measure trees, but the topology

preserves more of the tree-structure in limits, so that it is incomparable

to, and not coarser than, the standard topologies on metric measure trees.

With the example of the Aldous chain on cladograms, I also illustrate that

our new space can be very useful as state-space for stochastic processes in

order to obtain path-space diffusion limits of tree-valued Markov chains.

**Monday**, 8^{th} November 2021, 16:30, LMU, room B252, Theresienstr. 39, Munich

Hybrid meeting: Synchronous broadcast via zoom via the link (Click here for Zoom Meeting)

Sebastian Müller (Aix-Marseille Université)

Title: Voting-based probabilistic consensuses and their applications in distributed ledgers

Abstract: First, we give a short introduction in probabilistic consensus protocols and their applications in distributed ledgers. Then, we discuss the problem of metastability and how it can be resolved using common random numbers. After that, we present a current proposal, the so called On Tangle Voting (OTV), that uses the ledger itself as a voting layer. Finally, we conclude with first theoretical and numerical results, and several open questions. Joint work with the IOTA Foundation.

**Monday**, 15^{th} November 2021, 16:30, LMU, room B252, Theresienstr. 39, Munich

Online meeting via zoom: (Click here for Zoom Meeting)

Johannes Krebs (KU Eichstätt)

Title: Statistical topological data analysis

Abstract: We study selected statistical and probabilistic topics in persistent homology, which is the major branch of topological data analysis (TDA). TDA itself refers to a collection of statistical methods that find topological structure in data. Persistent homology is a multiscale approach to quantifying topological features in data, in particular, point cloud data.After a short and heuristic introduction to the main ideas of persistent homology, we will study multivariate and functional central limit theorems (CLT) and related stabilizing properties for persistent Betti numbers in the critical regime. Based on the multivariate CLT, we consider a smooth bootstrap procedure to construct confidence intervals. More- over, using a functional central limit theorem, we derive goodness-of-fit test in order to compare network structures for different types of underlying point processes.

**Monday**, 22^{th} November 2021, 16:30, LMU, room B252, Theresienstr. 39, Munich

Online meeting via zoom: (Click here for Zoom Meeting)

Cécile Mailler (University of Bath)

Title: A functional version of Kesten and Stigum's theorem

Abstract: If $(Z_n(\lambda))_{n\geq 0}$ is a discrete time Galton-Watson process with mean offspring $\lambda$, then $Z_n(\lambda)/\lambda^n$ is a non-negative martingale which converges almost surely to a limit $W(\lambda)$. In this joint work with JF Marckert, we define a process of Galton-Watson processes indexed by the mean offspring and show that the convergence of $\lambda\mapsto Z_n(\lambda)/\lambda^n$ to $\lambda\mapsto W(\lambda)$ holds in the space of càdlàg functions on [0,\infty), equipped with Skorokhod's topology on compact sets.

**Monday**, 6^{th} December 2021, 16:30, LMU, room B252, Theresienstr. 39, Munich

Hybrid meeting: Synchronous broadcast via zoom via the link (Click here for Zoom Meeting)

Guilherme Henrique de Paula Reis (TUM)

Title: The Ant Random Walk

Abstract: We propose a new model of random walk with reinforcement. The goal is to observe the ''Ant mill phenomenon'' in which a group of army ants are separated from the main group, lose the pheromone track and begin to follow one another, forming a continuously rotating circle. The ants are not able to go back home and will eventually die of exhaustion as we can see in the video "Why army ants get trapped in ‘death circles’" on youtube: https://www.youtube.com/watch?v=LEKwQxO4EZU . In this talk we introduce the Ant Random Walk which is a random walk with a special rule of reinforcement. We will see that this random walk exhibits the same phenomenon as the ''Ant mill phenomenon''.

Based in joint works with Dirk Erhard and Tertuliano Franco.

**Monday**, 20^{th} December 2021, 15:30, (using zoom)

ATTENTION: the lectures are NOT hybrid but purely online!

Lorenz Frühwirth (Universität Passau)

Title: The large deviation behaviour of lacunary sums

Abstract: In my talk I will present the main results from our recent article [2], where we gener-

alized some of the results in [1].

We study the large deviation behavior of lacunary sums (Sn/n)n∈Nwith Sn:=∑n

k=1 f (akU ), n ∈ N, where U is uniformly distributed on [0, 1], (ak)k∈Nis an

Hadamard gap sequence, and f : R→ Ris a 1-periodic, (Lipschitz-)continuous map-

ping. In the case of large gaps, we show that the normalized partial sums satisfy a large

deviation principle at speed n and with a good rate function which is the same as in

the case of independent and identically distributed random variables Uk, k ∈ N, hav-

ing uniform distribution on [0, 1]. When the lacunary sequence (ak)k∈Nis a geometric

progression, then we also obtain large deviation principles at speed n, but with a good

rate function that is different from the independent case, its form depending in a subtle

way on the interplay between the function f and the arithmetic properties of the gap

sequence.

**Im Anschluss daran:**

**Monday**, 20^{th} December 2021, 16:30, (using zoom)

Tom Kaufmann (Universität Bochum)

Title: Sharp Asymptotics for $q$-Norms of Random Vectors in High-Dimensional $\ell_p^n$-Balls

Abstract: Sharp large deviation results of Bahadur \& Ranga Rao-type are provided for the $q$-norm of random vectors distributed on the $\ell_p^n$-ball $\B^n_p$ according to the cone probability measure or the uniform distribution for $1 \le q<p < \infty$, thereby furthering previous large deviation results by Kabluchko, Prochno and Thäle in the same setting. These results are then applied to deduce sharp asymptotics for intersection volumes of different $\ell_p^n$-balls in the spirit of Schechtman and Schmuckenschläger. The sharp large deviation results are proven by providing convenient probabilistic representations of the $q$-norms, employing local limit theorems to approximate their densities, and then using geometric results for asymptotic expansions of Laplace integrals of Adriani \& Baldi and Liao \& Ramanan to integrate over the densities and derive concrete probability estimates.

arXiv-Link: https://arxiv.org/abs/2102.13513

**Monday**, 10^{th} January 2022, 16:30, (using zoom)

Sarah Penington (University of Bath)

Title: Branching random walk with non-local competition

Abstract: We study a particle system in which particles reproduce, move randomly in space, and compete with each other. We prove global survival as well as a shape theorem describing the asymptotic spread of the population, when the population density is sufficiently large. In contrast to most previous studies, we allow the competition kernel to have an arbitrary, or even infinite range, whence the term 'non-local competition'. This makes the particle system non-monotone and of infinite-range dependence, meaning that the usual comparison arguments break down and have to be replaced by a more hands-on approach. Based on joint work with Pascal Maillard.

**Monday**, 31^{th} January 2022, 16:30, (using zoom)

Christian Rehpenn (LMU; MSc presentation)

Title: Uniqueness of the infinite cluster for the weight-dependent random connection model

Abstract: We consider a large class of random graphs on the points of a Poisson point process in d dimensions, the weight-dependent random connection model. To each Poisson point we associate a random weight and we independently connect two points to each other with a probability depending on the distance and the weights of the two points. We investigate the number of infinite clusters and prove that the infinite cluster is almost surely unique. The general concept of the proof stems from the paper 'Uniqueness of the infinite component in a random graph with application to percolation and spin glasses' by Gandolfi et al. We extend their methods from the lattice to the continuum setup required for the weight-dependent random connection model.

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