Gilbert's disc model conditioned on the square lattice
arXiv:2607.14062
The paper defines a percolation model on the 2‑D integer lattice by placing a uniformly random point in each unit cell and connecting points whose Euclidean distance is below a radius R, and investigates the critical radii at which an infinite component appears and when the whole graph becomes connected.
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 . 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 . 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.