Tom Garcia-Sanchez

On the left is a simulation of the First Passage Percolation with Recovery process at time \(t=13\) with recovery rate \(\gamma=\frac{1}{2}\) on a Galton Waston tree with branching distribution \(\mathrm{Bin}(2,\frac{4}{5})\). The black edges show the infected cluster reached by the standard first-passage percolation process (with rate \(1\)) spreading from the root by time \(t\). Red vertices mark individuals whose recovery clocks (running at rate \(\gamma\)) were triggered upon infection by time \(t\), but had not yet rung. In [A1], we introduce and analyze this original model, where a recovery mechanism runs independently on top of a first-passage percolation process, deriving asymptotics for the size of red components as \(t\to\infty\).

The Directed Spanning Forest (DSF) of dimension \(d\geq 2\) is a random geometric graph whose vertex set is given by a homogeneous Poisson point process \(\mathcal N\) in \(\mathbb R^d\) and whose edges consist of all pairs \((x, y)\in\mathcal N^2\) such that \(y\) is the closest point to \(x\) in \(\mathcal N\) with a strictly larger \(e_d\) coordinate. The DSF has a natural forest structure: it is a collection of unrooted directed trees. A natural question is whether the DSF forms a single tree, meaning that all directed paths eventually coalesce, or whether it consists of several disjoint trees. In [P1], we address this question in arbitrary dimension obtaining that the DSF is almost surely a tree when \(d \leq 3\), and that it consists of infinitely many disjoint trees when \(d \geq 4\). On the right is a simulation of the 2-dimensional DSF in a finite box. In grey are represented its edges, while in red are highlighted the infinite paths starting from some vertices located below the bottom of the displayed area.

The Radial Spanning Tree (RST) of dimension \(d\geq 2\) is a random geometric graph constructed on a homogeneous Poisson point process \(\mathcal N\) in \(\mathbb R^d\) augmented by the origin, with edges connecting each \(x\in\mathcal N\) to the nearest point \(y\in\mathcal N\cup\{0\}\) that lies closer to 0 than \(x\), with respect to the Euclidean distance. By construction, it forms almost surely a tree rooted at 0. A natural question is whether infinite branches exists and are asymptotically directed. In [P2], we establish that the RST is almost surely straight in any dimension, meaning that the angular spread of its sub-trees vanishes as their sub-roots goes to infinity. This notion of straightness, introduced by Howard and Newman in 2001, allow to obtain that with probability one, all infinite branches are asymptotically directed, every possibility is attained, and directions reached by multiple infinite branches form a dense subset of \(\mathbb S^{d-1}\). On the left is a simulation of the 2-dimentional RST in a finite ball around the origin. Edges appear in light gray, with thickness decreasing with length. In red are highlighted some sub-trees exiting the displayed area, contained within translucent red cones pointing toward their sub-roots, with arbitrarily chosen apertures that appear to decrease to 0 with the distance to the origin, suggesting straightness. In black are highlighted some finite sub-trees remaining within the area, illustrating that RST sub-trees are not necessarily infinite.