arXiv subjects

David Geldbach

Publications and source records attributed to David Geldbach.

4 recordsLinked to original sources

arXiv2025

Hyperbolic branching Brownian motion: the empirical limit measure

We study branching Brownian motion in hyperbolic space. As hyperbolic Brownian motion is transient, the normalised empirical measure of branching Brownian motion converges to a random measure $\mu_\infty$ on the boundary. We show that the Hausdorff dimension of $ \mu_\infty$ is $(2\beta)\wedge 1$ where $\beta$ is the branching rate, and that $\mu_\infty$ admits a Lebesgue density for $\beta>1/2$. This is very different to the behaviour of the set of accumulation points on the boundary where $\beta_c=1/8$ which has been shown by Lalley and Sellke \cite{lalley_hyperbolic_1997}. This answers several questions posed by Woess \cite{woess_notes_2024} and similar questions posed by Candellero and Hutchcroft \cite{candellero_boundary_2023}. We believe that our methods also apply to branching random walks on non-elementary hyperbolic groups.

math.PR↗

arXiv2025

Polynomial slowdown in an angle-dependent 2d branching Brownian motion

We consider a branching Brownian motion in $\mathbb{R}^2$ in which particles independently diffuse as standard Brownian motions and branch at an inhomogeneous rate $b(\theta)$ which depends only on the angle $\theta$ of the particle. We assume that $b$ is maximal when $\theta=0$, which is the preferred direction for breeding. Furthermore we assume that $b(\theta ) = 1 - \beta \abs{\theta }^\alpha + O(\theta ^2)$, as $\theta \to 0$, for $\alpha \in (2/3,2)$ and $\beta>0.$ We show that if $M_t$ is the maximum distance to the origin at time $t$, then $(M_t-m(t))_{t\ge 1}$ is tight where $$m(t) = \sqrt{2} t - \frac{\vartheta_1}{\sqrt{2}} t^{(2-\alpha)/(2+\alpha)} - \left(\frac{3}{2\sqrt{2}} - \frac{\alpha}{2\sqrt{2}(2+\alpha)}\right) \log t. $$ and $\vartheta_1$ is explicit in terms of the first eigenvalue of a certain operator.

math.PR↗

arXiv2025

Biased branching random walks on Bienaym\'e--Galton--Watson trees

We study $\lambda$-biased branching random walks on Bienaym\'e--Galton--Watson trees in discrete time. We consider the maximal displacement at time $n$, $\max_{\vert u \vert =n} \vert X(u) \vert$, and show that it almost surely grows at a deterministic, linear speed. We characterize this speed with the help of the large deviation rate function of the $\lambda$-biased random walk of a single particle. A similar result is given for the minimal displacement at time $n$, $\min_{\vert u \vert =n} \vert X(u) \vert$.

math.PR↗

arXiv2023

Continuum asymptotics for tree growth models with uniform backward dynamics

We study (plane) tree-valued Markov chains $(T_n,n \geq 1)$ with uniform backward dynamics and show that they can be obtained by sampling from a real tree. As non--plane trees, every such Markov chain is represented by a weighted real tree. We equip this real tree with a planar order as well as some extra functions for the full representation theorem. We also show that under an inhomogeneous rescaling after trimming leaves $(T_n, n\geq 1)$ converges to a random real tree in the Gromov--Prokhorov metric. This makes use of a special class of real trees, interval partition trees, which were introduced by Forman (2020). Moreover, this generalises and sheds some new light on work by Evans, Gr\"ubel and Wakolbinger (2017) on the binary special case.

math.PR↗