Take a random configuration of $n$ non-overlapping discs of radius $r$ in the unit square $[0,1]^2$. (You could think of this as taking $n$ points uniform randomly in $[r,1-r]^2$ and then restricting attention to the case that every pair of points $\{ x,y \}$ satisfies $d(x,y) \ge 2r$.)

From the point of view of statistical mechanics, the interesting case is when $r^2 n \to C$ for some constant $C > 0 $ as $n \to \infty$, and various kinds of phase transitions have been studied experimentally, but little seems known mathematically.

What I am wondering about is whether anyone has studied the following kind of percolation. Set $\lambda >1$ to be a fixed parameter to measure proximity. Define a graph by considering the centers of the $n$ discs to be the vertices, and connecting a pair $\{ x,y \}$ by an edge whenever $d(x,y) < 2 \lambda$.

My question is: given some choice of $\lambda$ does there exist a critical threshold $C_t$ (depending on $\lambda$) such that whenever $C > < C_t$ all the connected components of this graph are likely to be small, of order $O(\log n)$ or even $o(n)$, and whenever $C > C_t$ there is a giant component, of order $\Omega(n)$?

What I know about is that for geometric random graphs on i.i.d. random points, percolation is known to occur for fairly general distibutions, and that this is closely related to bond percolation on a lattice. But in the hard spheres distibution points are far from being independent.

I would also be interested to hear about percolation on other kinds of repulsive point processes -- Matern, Strauss, etc.

Continuum Percolationby Meester and Roy, or the continuum percolation chapter in Grimmett'sPercolation. $\endgroup$