arXiv subjects

Georg Braun

Publications and source records attributed to Georg Braun.

3 recordsLinked to original sources

arXiv2021

Boolean percolation on digraphs and random exchange processes

We study, in a general graph-theoretic formulation, a long-range percolation model introduced by Lamperti. For various underlying directed graphs, we discuss connections between this model and random exchange processes. We clarify, for $n \in \mathbb{N}$, under which conditions the lattices $\mathbb{N}_0^n$ and $\mathbb{Z}^n$ are essentially covered in this model. Moreover, for all $n \geq 2$, we establish that it is impossible to cover the directed $n$-ary tree in our model.

math.PR↗

arXiv2020

On Supercritical Branching Processes with Emigration

We study supercritical branching processes under the influence of an i.i.d. emigration component. We provide conditions, under which the lifetime of the process is finite respectively has a finite expectation. A new version of the Kesten-Stigum theorem is obtained and the extinction probability for a large initial population size is related to the tail behaviour of the emigration.

math.PR↗

arXiv2020

On the Growth of a Ballistic Deposition Model on Finite Graphs

We revisit a ballistic deposition process introduced by Atar, Athreya and Kang. Let $\mathcal{G}=(V,E)$ be a finite connected graph. We choose independently and uniformly vertices in $\mathcal{G}$. If a vertex $x$ is chosen and the previous height configuration is given by $h=(h_y)_{y \in V} \in \mathbb{N}_0^V$, the height $h_x$ is replaced by \[ \tilde{h}_x := 1 + \max_{y \sim x} h_y. \] We study asymptotic properties of this growth model. We determine the asymptotic growth parameter $γ(\mathcal{G} )$ for some graphs and prove a central limit theorem for the fluctuations around $γ( \mathcal{G})$. We also give a new graph-theoretic interpretation of an inequality obtained by Atar et al..

math.PR↗