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). > <b>Q1</b>. 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: <br /> <img src="http://cs.smith.edu/~orourke/MathOverflow/RandCircleLinks.jpg" alt="RandCircleLinks" /> <br /> > <b>Q2</b>. Same as <b>Q1</b>, but with $S$ a sphere of some (perhaps large) radius $r > 1$. > <b>Q3</b>. Same as <b>Q1</b>, except with $S$ an arbitrary convex body, e.g., a cube. I feel the answer to <b>Q1</b> should be *Yes*, but I am less certain of <b>Q3</b>. 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! <b>Answered</b> (*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!