Gilbert's disc model conditioned on the square lattice

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.