Gilbert's disc model conditioned on the square lattice
2026-07-15 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors introduce a new way to connect points randomly placed inside each square of an infinite grid by linking points closer than a certain distance. They study the smallest distance needed so that an endless group of connected points appears almost always. They also look at two special distances: one where it's possible to arrange points to get an infinite group, and one where every point is connected. This helps understand how connectivity changes with distance in this grid-based random setup.
percolation theorytwo-dimensional latticecontinuous percolationrandom geometric graphconnected componentcritical radiusinfinite clustergraph connectivityEuclidean distancerandom point process
Authors
Jérôme Casse, Irène Marcovici, Maxence Poutrel
Abstract
We present a new percolation model on the two-dimensional lattice, which can be seen as a conditioned version of continuous percolation on the plane. Let us place a point uniformly at random in each cell of the grid $\mathbb{Z}^2$. These points correspond to the vertices of our graph, and we connect two points by an edge if their distance is less than a fixed radius $R$. We are interested in the radius from which there exists almost surely an infinite connected component. We also study two other critical radii specific to the geometry of our model: the smallest radius such that there exists a positioning of the points for which there is an infinite connected component, and the radius from which all points are connected to each other.