On a uniformly random chord diagram and its intersection graph

Abstract

A chord diagram refers to a set of chords with distinct endpoints on a circle. The intersection graph of a chord diagram C is defined by substituting the chords of C with vertices and by adding edges between two vertices whenever the corresponding two chords cross each other. Let Cn and Gn denote the chord diagram chosen uniformly at random from all chord diagrams with n chords and the corresponding intersection graph, respectively. We analyze Cn and Gn as n tends to infinity. In particular, we study the degree of a random vertex in Gn, the k-core of Gn, and the number of strong components of the directed graph obtained from Gn 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.