Context
We consider the plane, $\mathbb{R}^2$, which we split into squares of size $1$. Each square has a probability $p$ of being black and $1-p$ of being white, independently for each square. Let $F$ be the area covered by the black squares; this is a random subset of $\mathbb{R}^2$.
For description, we say that a set $K \subset F$ is connected if for any pair of squares in $K$ there exists a path of touching black squares in $K$ connecting them. Obviously, $F$ is a union of separate connected components, which are called clusters.
It is well known (see for example Percolation Theory for Mathematicians. Birkhauser, Basel, Boston, 1982.
Khruslov, E.Ya.) that there exists a threshold probability $p_c$ such that
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for $p<p_c$, $F$ is made of finite clusters (a.s.)
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for $p>p_c$, $F$ contains an infinite cluster (a.s.),
and it was proved that $p_c \approx 0.41.$ This phenomenon is known as percolation.
My question
I would like to study a similar problem in $\mathbb{R}^2$ but a little bit different. In my case, I have a selection of spheres $B_i=B(x_i,1),\ i \in \mathbb{N}$, where the family of the sphere's center $(x_i)_{i \in \mathbb{N}}$ is made of random variables (see commentary, for example it could be a random Poisson Hardcore Point Process) valued in $\mathbb{R}^2$ and such as $B_i \cap B_j = \oslash,\ i \neq j$. The spheres $F=\cup_{i \in \mathbb{N}} B_i$ are colored in black and the rest $\mathbb{R}^2 \setminus \cup_{i \in \mathbb{N}} B_i$ is colored in white.
As with the random checkerboard studied above, it is possible to define the notion of a cluster in $F$ for touching spheres, and it is possible that an infinite cluster of spheres appears.
I was wondering if there were references about an equivalent of this probability threshold ? I would especially like a reference on this topic if it exists.
(Feel free to retag this question as appropriate.)
Best Answer
I have to go with Steven Stadnicki's comment and say that your distribution for the point process has to be very weird for the spheres to be non-intersecting, yet touching each other often enough to build large clusters. I do not know if this is helpful in your setting, but maybe you should look into percolation theory on more general geometries, for instance on isoradial graphs.