# Probability Seminar Essen

## Probability Seminar Essen

Covers a wide range of topics in Probability and its applications.

 Apr 16 Viktor Schulmann, TU Dortmund Apr 23 Sascha Kissel, University Bochum May 7 Gabriel Hernán Berzunza Ojeda, University Uppsala May 14 Jan Nagel, University Dortmund May 21 José Manuel Pedraza, London School of Economics and Political Science May 28 Carina Betken, University Bochum Jun 4 Jiří Cerny, University Basel Jun 25 David Belius, University Basel Jul 2 Jul 9 Max Grieshammer, University Erlangen

## Abstracts:

 Apr 16 Viktor Schulmann, TU Dortmund Life span estimation for randomly moving particles based on their places of death Consider the following problem from physics: A radiation source is placed at the center of a screen. At certain time intervals the source releases particles. These move around the screen following a path of some known random process $(Y_t)_{t\geq 0}$ without interacting with each other and without us being able to observe their movement until they die after some random time $T$. During its death a particle leaves a mark such that we can measure the distance $X=||Y_T||_2$ it traveled from the source during its lifetime. Based on these observed distances we wish to infer the life span $T$ of a particle or, in particular, the density $f_T$ of $T$. We assume $(Y_t)_{t\geq 0}$ from our physics experiment to be a multi-dimensional L\'{e}vy processes with spherical symmetry. Norms of such processes exhibit structural similarities to one-dimensional L{\'e}vy processes. For that case an estimator was given by Belomestny and Schoenmakers (2016) using the Mellin and Laplace transforms. Applying their techniques we construct a non-parametrical estimator for $f_T$, calculate its convergence rate and show its optimality in the minimax sense. Apr 23 Sascha Kissel, University Bochum Dynamical Gibbs-non-Gibbs transitions in Curie-Weiss Widom-Rowlinson models In this talk, we consider the Curie-Weiss Widom-Rowlinson model for particles with spins and holes, with a repulsion strength $\beta>0$ between particles of opposite spins. A closed solution of the symmetric model will be provided. After this, we talk about the dynamical Gibbs-non-Gibbs transitions for the time-evolved model under independent stochastic symmetric spin-flip dynamics. It will be shown that, for sufficiently large $\beta$ after a transition time, continuously many bad empirical measures appear. May 7 Gabriel Hernán Berzunza Ojeda, University Uppsala Fluctuations of the giant cluster for supercritical percolation on split trees A split tree of cardinality $n$ is constructed by distributing $n$ "balls" (which often represent "key numbers") in a subset of vertices of an infinite tree. In this talk, we will discuss Bernoulli bond percolation on arbitrary split trees of large but finite cardinality $n$. The main goal is to show for appropriate percolation regimes that there exists a unique giant cluster that is of a size comparable of that of the entire tree (where size is defined as either the number of vertices or the number of balls). Furthermore, it will be shown that in such percolation regimes, (also known as supercritical regimes) the fluctuations of the size of the giant cluster are non-Gaussian as $n$ grows. Instead, they are described by an infinitely divisible distribution that belongs to the class of stable Cauchy laws. This work is a generalization of previous results for random $m$-ary recursive trees which is one specific case of split trees. Other important examples of split trees include $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees, fringe-balanced trees, digital search trees and random simplex trees. The approach is based on a remarkable decomposition of the size of the giant percolation cluster as a sum of essentially independent random variables which allows us to apply a classical limit theorem for the convergence of triangular arrays to infinitely divisible distributions. May 14 Jan Nagel, University Dortmund Random walk on barely supercritical branching random walk The motivating question behind this project is how a random walk behaves on a barely supercritical percolation cluster, that is, an infinite percolation cluster when the percolation probability is close to the critical value. As a more tractable model, we approximate the percolation cluster by the embedding of a Galton-Watson tree into the lattice. When the random walk runs on the tree, the embedded process is a random walk on a branching random walk. Now we can consider a barely supercritical branching process conditioned on survival, with survival probability approaching zero. In this setting the tree structure allows a fine analysis of the random walk and we can prove a scaling limit for the embedded process under a nonstandard scaling. The talk is based on a joint work with Remco van der Hofstad and Tim Hulshof. May 21 José Manuel Pedraza, London School of Economics and Political Science Predicting in a $L_p$ sense the last zero of an spectrally negative Lévy process Given a spectrally negative Lévy process drifting to infinity, we are interested in the last time g in which the process is below zero. At any time t, the value of g is unknown and it is only with the realisation of the whole process when we can know when the last zero of the process occurred. However, this is often too late, we usually are interested in know how close is the process to g at time t and take some actions based on this information. We are interested on finding a stopping time which is as close as possible to $g$ (on a $L_p$ distance). We prove that solving this optimal prediction problem is equivalent to solve an optimal stopping problem in terms of a two dimensional Markov process which involves the time of the current excursion away from the negative half line and the L\'evy process. We state some basic properties of the last zero process and prove the existence of the solution of the optimal stopping problem. Then we show the solution of the optimal stopping problem (and therefore the optimal prediction problem) is given as the first time that the process crosses above a non-increasing and non-negative curve dependent on the time of the last excursion away from the negative half line. May 28 Carina Betken, University Bochum      Stein's method and preferential attachment random graphs We consider a general preferential attachment model, where the probability that a newly arriving vertex  connects to an older vertex is proportional to a (sub-)linear function  of the indegree of the older vertex at that time. We develop Stein's method for the asymptotic indegree distribution of a vertex, chosen uniformly at random, and deduce rates of convergence as the number of vertices tends to $\infty$. Using Stein's method for Poisson and  Normal approximation we also show limit theorems for the outdegree distribution as well as for the number of isolated vertices. June 4 Jiří Cerny, University Basel   Gaussian free field on regular graphs We study the behaviour of level sets of 0-mean Gaussian free field on regular expanding graphs, and (at least partially) prove that they exhibit a similar phase transition as Bernoulli percolation and the vacant set of random walk on such graphs. Jun 25 David Belius, University Basel   The TAP approach to mean field spin glasses The topic of this talk is mean-field spin glasses, in particular the Sherrington-Kirkpatrick (SK) model. I will revisit the Thouless-Andersson-Palmer approach to the these models from the physics literature, and report on our efforts to use it as a basis for an alternative mathematically rigorous treatment. Jul 9 Max Grieshammer, University Erlangen Measure representation of evolving genealogies