Formation of a giant component in the intersection graph of a random chord diagram

TL;DR

This paper analyzes the phase transition in the largest component of a random chord diagram's intersection graph, revealing how component size depends on the ratio of crossings to chords.

Contribution

It provides a new asymptotic analysis of the largest component in random chord diagrams using advanced combinatorial formulas.

Findings

For m/nlog(n) in (0,2/π^2), the largest component contains almost all crossings and a positive fraction of chords.

When m < n/14, the largest component is of size O(log n).

Derived asymptotic formulas for the number of chord diagrams with given parameters.

Abstract

We study the number of chords and the number of crossings in the largest component of a random chord diagram when the chords are sparsely crossing. This is equivalent to studying the number of vertices and the number of edges in the largest component of the random intersection graph. Denoting the number of chords by n and the number of crossings by m, when m/nlog(n) tends to a limit in (0,2/\pi^2), we show that the chord diagram chosen uniformly at random from all the diagrams with given parameters has a component containing almost all the crossings and a positive fraction of chords. On the other hand, when m < n/14, the size of the largest component is of size O(log n). One of the key analytical ingredients is an asymptotic expression for the number of chord diagrams with parameters n and m for m <(2/\pi^2)n\log(n), based on the Touchard-Riordan formula and the Jacobi identity for Euler partition function.