TITLE: Random rings linked into one component?
QUESTION [17 upvotes]: Let $S$ be a sphere of unit radius.
Let $C_n$ be a collection of unit-radius circles/rings whose centers
are (uniformly distributed)
random points in $S$, and which are oriented (tilted) randomly (again, uniformly).

Q1.
  As $n \to \infty$, does the probability that all the
  rings in $C_n$ are linked together in one component approach $1$?

By "linked together" I mean that if you pick up any one ring,
all the others are physically connected and would follow.
For example, below there are $n=5$ rings, four of which are
connected, but one (topmost) is not:

          



Q2.  Same as Q1, but with $S$ a sphere of some (perhaps large) radius
  $r > 1$.
Q3.  Same as Q1, except with $S$ an arbitrary convex
  body, e.g., a cube.

I feel the answer to Q1 should be Yes, but I am
less certain of Q3.
Exact computation of probabilities as a function of $n$ might be difficult,
but I am hoping there are relatively simple arguments to settle these
questions. Thanks for ideas!
Answered (1May13).
The combination of
Ori Gurel-Gurevich's
and
Benoît Kloeckner's postings constitute a rather complete answer,
establishing that the answer to all my questions is Yes, even
without the assumption that $S$ is convex.
Thanks for the interest!

REPLY [5 votes]: I think I can complement the answer of Ori Gurel-Gurevich to prove that indeed, when we deal with connected open sets (no need for convexity) the answer is positive.
1. There is a finite configuration of circles $C_1, \dots, C_N$ whose centers are in the domain $D$, such that any circle $C$ with center in the domain $D$ must be linked to at least one of the $C_i$.
One way to do this is to first take a very tight lattice $\Lambda$, and put horizontal circles $C_1,\dots, C_K$ with centers on $\Lambda\cap D$. Now, a circle with center in $D$ that is not linked to any of these $C_i$ ($i\leq K$) must be roughly horizontal.
Then, add circles $C_{K+1},\dots, C_N$ with center on the lattice but oriented along a given vertical plane. A circle not linked to any $C_i$ must be roughly horizontal and roughly vertical, thus don't exist.
2. The above construction is stable under small perturbation. This means that there are small open sets $U_1,\dots, U_N$ of the parameter space such that for all set of circles $C_1,\dots, C_N, C$ such that $C_i\in U_i$, $C$ must be linked to one of the $C_i$.
3. By adding more circles that link together the $C_i$ of 1., we could have assumed that the $C_i$ make a linked component (this is where we use the connectedness assumption on $D$, which in fact could be weakened). As in 2., this is stable under small perturbation, so in fact there are small open sets $U_1,\dots, U_N$ of the parameter space such that for all set of circles $C_1,\dots, C_N, C$ such that $C_i\in U_i$, the $N+1$ circles must be linked together.
4. Now, when $n\to \infty$ the probability that circles have been drawn in each of the $U_i$ increase to $1$ exponentially fast, so at some point our random configuration contains with high probability a set of circles that links all admissible  circles, including all the other randomly drawn ones.