arXiv subjects
Publications and source records attributed to Julian Kern.
A full solution to the recently proposed problem of determining the probability that no $k$-gon can be built from $n$ independently and uniformly chosen sticks in $[0,1]$ is proposed. This extends the known results for triangles and quadrilaterals to general $k$-gons and offers a clearer interpretation of the connection to products of $k$-bonacci numbers.
We present a model for growth in a multi-species population. We consider two types evolving as a logistic branching process with mutation, where one of the types has a selective advantage, and are interested in the regime in which the carrying capacity of the system goes to $\infty$. We first study the genealogy of the population up until it almost reaches carrying capacity through a coupling with an independent branching process. We then focus on the phase in which the population has reached carrying capacity. After recovering a Gillespie--Wright--Fisher SDE in the infinite carrying capacity limit, we construct the Ancestral Selection Graph and show the convergence of the lineage counting process to the moment dual of the limiting diffusion.
We consider a class of generalized long-range exclusion processes evolving either on $\mathbb Z$ or on a finite lattice with an open boundary. The jump rates are given in terms of a general kernel depending on both the departure and destination sites, and it is such that the particle displacement has an infinite expectation, but some tail bounds are satisfied. We study the superballisitic scaling limit of the particle density and prove that its space-time evolution is concentrated on the set of weak solutions to a non-local transport equation. Since the stationary states of the dynamics are unknown, we develop a new approach to such a limit relying only on the algebraic structure of the Markovian generator.
We propose an extension of the classical $\Lambda$-Fleming-Viot model to intrinsically varying population sizes. During events, instead of replacing a proportion of the population, a random mass dies and a, possibly different, random mass of new individuals is added. The model can also incorporate a drift term, representing infinitesimally small, but frequent events. We investigate elementary properties of the model, analyse its relation to the $\Lambda$-Fleming-Viot model and describe a duality relationship. Through the lookdown framework, we provide a forward-in-time analysis of fixation and coming down from infinity. Furthermore, we present a new duality argument allowing one to deduce well-posedness of the measure-valued process without the necessity of proving uniqueness of the associated lookdown martingale problem.
We provide a short proof for the exponential equivalence between misanthrope processes in contact with weak reservoirs and those with impermeable boundaries. As a consequence, we can derive both the hydrodynamic limit and the large deviations of the totally asymmetric simple exclusion process (TASEP) in contact with weak reservoirs. This extends a recent result which proved the hydrodynamic behaviour of a vanishing viscocity approximation of the TASEP in contact with weak reservoirs. Further applications to a class of asymmetric exclusion processes with long jumps are discussed.
We consider a random diffusion dynamics for an infinite system of hard spheres of two different sizes evolving in $\mathbb{R}^d$, its reversible probability measure, and its projection on the subset of the large spheres. The main feature is the occurrence of an attractive short-range dynamical interaction -- known in the physics literature as a depletion interaction -- between the large spheres, which is induced by the hidden presence of the small ones. By considering the asymptotic limit for such a system when the density of the particles is high, we also obtain a constructive dynamical approach to the famous discrete geometry problem of maximisation of the contact number of $n$ identical spheres in $\mathbb{R}^d$. As support material, we propose numerical simulations in the form of movies.
This paper presents a gentle and informal introduction to the Skorokhod topologies. Focus is on motivating examples and concepts.
In 2017, Duminil-Copin et al. introduced the OSSS method to study properties of diverse percolation models. This document aims to introduce the reader to this new method. It contains a introduction to percolation theory, then concentrates on the case of Poisson-Boolean percolation. The majority of this document is dedicated to an detailed analysis of [DRT18]. This work is the result of an internship in Summer 2019 with Jean-Baptiste Gouéré at the University of Tours.
This thesis examins a generalisation of polar decompositions to indefinite inner product spaces. The necessary general theory is studied and some general results are given. The main part of the thesis focuses on polar decompositions with commuting factors: First, a proof for a generalisation of the link between polar decomposition with commuting factors and normal matrices is given. Then, some properties of such decompositions are studied and it is shown that the commutativity of the factors only depends on the selfadjoint part. Eventually, polar decompositions with commuting factors are studied under similarity transformations that do not alter the structure of the space. For this purpose, normal forms are decomposed and analysed.