Research

Articles and preprints

Hyperbolic branching Brownian motion: the empirical limit measure arXiv [2509.06730]

Abstract: 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 $\mathrm{supp }\mu_\infty$ 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 Selke. This answers several questions posed by Woess in a recent survey article and similar questions posed by Candellero and Hutchcroft. We believe that our methods also apply to branching random walks on non-elementary hyperbolic groups.

Polynomial slowdown in space-inhomogeneous branching Brownian motion arXiv [2506.10623]

(Joint work with Julien Berestycki and Michel Pain)

Abstract: 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 $b(\theta ) = 1 - \beta \vert\theta \vert^\alpha + O(\theta ^2)$, 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 $(M_t-m(t))_{t\geq 1}$ 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.

Biased branching random walks on Bienaymé-Galton-Watson trees arXiv [2502.07363]

(Joint work with Julien Berestycki, Nina Gantert and Quan Shi)

Abstract: We study $\lambda$-biased branching random walks on Bienaymé-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 $n$, $\min_{\vert u \vert =n} \vert X(u)\vert$.

Continuum asymptotics for tree growth models arXiv [2309.04336]

Abstract: We classify the forward dynamics of all (plane) tree-valued Markov chains $(T_n,n \geq 1)$ with uniform backward dynamics. Every such Markov chain is classified by a weighted real tree, decorated with some additional functions. 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 generalises and sheds some new light on work by Evans, Grübel and Wakolbinger (2017) on the binary special case.

Old theses

Here are my old theses as pdfs:

Bachelor thesis: The Scaling Limit of the Erdös-Rény Graph

Master thesis: Ergodicity of the dynamical XY-model