  A chord diagram refers to a set of chords with distinct endpoints on a
circle. The intersection graph of a chord diagram $\cal C$ is defined by
substituting the chords of $\cal C$ with vertices and by adding edges between
two vertices whenever the corresponding two chords cross each other. Let $C_n$
and $G_n$ denote the chord diagram chosen uniformly at random from all chord
diagrams with $n$ chords and the corresponding intersection graph,
respectively. We analyze $C_n$ and $G_n$ as $n$ tends to infinity. In
particular, we study the degree of a random vertex in $G_n$, the $k$-core of
$G_n$, and the number of strong components of the directed graph obtained from
$G_n$ by orienting edges by flipping a fair coin for each edge. We also give
two equivalent evolutions of a random chord diagram and show that, with
probability approaching $1$, a chord diagram produced after $m$ steps of these
evolutions becomes monolithic as $m$ tends to infinity and stays monolithic
afterward forever.
