TITLE: Is there a percolation threshold in the hard discs model?
QUESTION [11 upvotes]: 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.

REPLY [4 votes]: Perhaps you are already aware of asymptotics for component sizes in the continuum percolation model. It is there in the book of M. Penrose.
Since you asked about percolation on repulsive point processes, here is a compressed version of our results of non-trivial phase transition on sub-Poisson point processes , point processes that are less clustered than a Poisson in a certain sense. Our main example is a perturbed lattice. We also show that stationary determinantal point processes have a non-zero critical intensity as regards percolation. I think these point processes can be considered as repulsive point processes.