IS07: Inhomogeneous spatial graph models

date: 7/14/2025, time: 14:00-15:30, room: ICS 25

Organizer: Júlia Komjáthy (Delft University of Technology)

Chair: Júlia Komjáthy (Delft University of Technology)

Crossing probabilities in geometric inhomogeneous random graphs

Emmanuel JACOB (ENS Lyon)

In a geometric inhomogeneous random graph vertices are given by the points of a Poisson process and are equipped with independent weights following a heavy tailed distribution. Any pair of distinct vertices is independently forming an edge with a probability decaying as a function of the product of the weights divided by the distance of the vertices. For this continuum percolation model we study the probability of existence of paths crossing annuli with increasing inner and outer radii in the quantitatively subcritical phase. Depending on the inner and outer radius of the annulus, the power-law exponent of the degree distribution and the decay of the probability of long edges, we identify regimes where the crossing probabilities by a path are equivalent to the crossing probabilities by one or by two edges. As a corollary we get the subcritical one-arm exponents characterising the decay of the probability that a typical point is in a component with a large diameter.

High-intensity Voronoi percolation on manifolds

Barbara Dembin (CNRS, Université de Strasbourg)

Consider a \(d\)-dimensional complete Riemannian manifolds (for simplicity one can just think of \(\mathbb{H}^d\) for \(d \geq 2\)). In this setting, Voronoi cells are constructed from a homogeneous Poisson point process of intensity \(\lambda\), and each cell is independently colored white with probability \(p\) and black with probability \(1-p\). We focus on the percolation properties of the resulting random coloring as the intensity \(\lambda\) goes to infinity. Our main result shows that, under mild geometric assumptions on the manifold \(M\), both the critical percolation threshold \(p_c(M,\lambda)\) for the emergence of an unbounded white cluster and the uniqueness threshold \(p_u(M,\lambda)\) converge to the Euclidean critical threshold \(p_c(\mathbb{R}^d)\) in the high-intensity limit \(\lambda \to \infty\).

Truncating the marked random connection model

Johannes Bäumler (University of California, Los Angeles)

In the marked random connection model, for each pair of points \(\{ x, y \}\), there is an open edge between these points with probability depending on the Euclidean distance between the points and the marks of the points. When are the long edges necessary for the existence of an infinite cluster? The truncation problem asks whether one can remove all long enough edges while still retaining an infinite open cluster. We discuss this question in the strong decay regime.