arXiv subjects

Tom Garcia-Sanchez

Publications and source records attributed to Tom Garcia-Sanchez.

4 recordsLinked to original sources

arXiv2026

Connected components and topological ends of stationary planar forests

We study the topological structure of random geometric forests $G$ in the Euclidean plane under mild assumptions: non-crossing edges, stationarity, and finite edge intensity. The framework covers a broad range of constructions, including models based on stationary point processes as well as lattices, and encompasses many already well-studied examples among drainage networks, geodesic forests arising from first- and last-passage percolation, and minimal or uniform spanning trees. First, denoting by $N_k$ the number of $k$-ended connected components in $G$ for each $k\geq0$, we show that almost surely, all trees of $G$ have at most two topological ends, $N_0\in\{0,\infty\}$, $N_1\leq2$, and $N_1=2\implies N_2<\infty$. We then construct explicit examples realizing all possibilities compatible with these constraints, yielding a complete classification of the admissible topological structures for $G$. As a second result, we prove that under the additional assumptions that $G$ is non-empty, oriented, out-degree one, with all its directed paths going to infinity along a fixed deterministic direction, the situation reduces to a dichotomy: $G$ consists almost surely of either a unique one-ended tree, or infinitely many two-ended trees. The latter extends a theorem of Chaika and Krishnan (2019), who considered a lattice setting. Our proofs combine classical Burton-Keane type arguments with substantial new conceptual ideas using planar topology, resulting in a robust, unified approach.

math.PR↗

arXiv2026

The Radial Spanning Tree is straight in all dimensions

The Radial Spanning Tree (RST) in dimension $d\geq2$ 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 $0$. The RST was introduced in 2007 by Baccelli and Bordenave, who investigated straightness, a deterministic property introduced by Howard and Newman in 2001, to derive information about the asymptotic directions of infinite branches. They proved that the RST is almost surely straight in dimension $2$, which directly implies that all infinite branches are asymptotically directed, every possibility is attained, and directions reached by multiple infinite branches form a dense subset. However, their approach relies crucially on planarity, preventing any straightforward extension to higher dimensions. In this paper, we close this gap by proving that the RST is almost surely \emph{straight} in any dimension, thereby obtaining the same consequences for the behavior of infinite branches. Our approach resolves the key barriers in the study of the RST, notably those posed by the complex dependency structure combined with the radial nature of the model, and especially beyond the planar setting. It relies on tools developed for the analysis of the Directed Spanning Forest, a closely related model, including recent progress by the author in 2025. Specifically, a key contribution of this work is the construction of a suitable renewal-type decomposition of RST paths. Leveraging this decomposition together with classical concentration inequalities, we show that RST paths cannot deviate far from straight lines and derive straightness.

math.PR↗

arXiv2025

The Directed Spanning Forest: coalescence versus dimension

For $p\in[1,\infty]$, the $\ell^p$ directed spanning forest (DSF) of dimension $d\geq 2$ is an oriented random geometric graph whose vertex set is given by a homogeneous Poisson point process $\mathcal N$ on $\mathbb R^d$ and whose edges consist of all pairs $(x, y)\in\mathcal N^2$ such that $y$ is the closest point to $x$ in $\mathcal N$ for the $\ell^p$ distance among points with a strictly larger $e_d$ coordinate. First introduced by Baccelli and Bordenave in 2007 in the case $p=d=2$, this graph has a natural forest structure. In this work, we study the number of disjoint trees in the $\ell^p$ DSF for arbitrary dimensions $d\geq2$ and various values of $p\in [1,\infty]$. We prove that for $p\in\{1, 2,\infty\}$, the graph is almost surely a tree when $d=3$, and consists of infinitely many disjoint trees when $d\geq 4$. Additionally, we show that for all $p\in[1,\infty]$, the DSF in dimension $2$ is almost surely a tree and, under appropriate diffusive scaling, converges weakly to the Brownian web, generalizing the result previously known for p=2. Although these results were expected from a heuristic point of view, and the main strategies and tools were largely understood, extending them beyond the planar setting ($d=2$) and to the singular case $p=\infty$ presented a significant challenge. Notably, in the absence of planarity, which plays a crucial role in existing arguments, delicate and innovative techniques were required to manage the complex geometric dependencies of the model. We develop substantially new ideas to handle arbitrary dimension $d\geq2$ and various values of $p\in [1,\infty]$ within a unified framework. In particular, we introduce a novel stochastic domination argument that allows us to compare the fully dependent model with a simplified version in which the geometric correlations are suppressed.

math.PR↗

arXiv2024

First Passage Percolation with Recovery

First passage percolation with recovery is a process aimed at modeling the spread of epidemics. On a graph $G$ place a red particle at a reference vertex $o$ and colorless particles (seeds) at all other vertices. The red particle starts spreading a \emph{red first passage percolation} of rate $1$, while all seeds are dormant. As soon as a seed is reached by the process, it turns red and starts spreading {red first passage percolation}. All vertices are equipped with independent exponential clocks ringing at rate $γ>0$, when a clock rings the corresponding \emph{red vertex turns black}. For $t\geq 0$, let $H_t$ and $M_t$ denote the size of the longest red path and of the largest red cluster present at time $t$. %, respectively. If $G$ is the semi-line, then for all $γ>0$ almost surely $\limsup_{t}\frac{H_t\log\log t}{\log t}=1 $ and $\liminf_{t}H_t=0$. In contrast, if $G$ is an infinite Galton-Watson tree with offspring mean $\mathbf{m}>1$ then, for all $γ>0$, almost surely $\liminf_{t}\frac{H_t\log t}{t}\geq\mathbf{m}-1 $ and $\liminf_{t}\frac{M_t\log\log t}{t}\geq \mathbf{m}-1$, while $\limsup_{t} \frac{M_t}{e^{c t}}\leq 1$, for all $c>\mathbf{m} -1$. Also, almost surely as $t\to \infty$, for all $γ>0$ $H_t$ is of order at most $t$. Furthermore, if we restrict our attention to bounded-degree graphs, then for any $\varepsilon>0$ there is a critical value $γ_c>0$ so that for all $γ>γ_c$, almost surely $\limsup_{t}\frac{M_t}{t}\leq \varepsilon $.

math.PR↗