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

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/n log n tends to a limit in (0, 2/π 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/π 2 )n log n, based on the Touchard-Riordan formula and the Jacobi identity for the generating function of Euler partition function.

The Brownian limit of separable permutations

“…We also refer to Acan and Pittel [AP13] for an analysis of inversion graphs of uniform random permutations with a fixed number of inversions (thus fixing the number of edges). • In a similar spirit, Acan has studied various properties of the intersection graph of a uniform random chord diagram in [Aca17]; see also Acan and Pittel [AP17] for an analysis of intersection graphs of uniform random chord diagrams with a fixed number of crossings (fixing again the number of edges in the graph). • The graphon limit of a uniform random string graph has been considered by Janson and Uzzell [JU17], who identified a set of possible limit points, and conjectured the actual graphon limit.…”

On a memory game and preferential attachment graphs

“…pairings of 2n points. (LCDs appear in various fields in mathematics; see, for example, [2], [3], [8]- [11], [17], and [19].) First, the connection between the memory game and LCDs is made by noting that each game is equivalent to a unique game in which the second occurrence of card i precedes the second occurrence of card j for i < j.…”

Convergence rates of limit theorems in random chord diagrams

“…Additionally they showed that with high probability, a random chord diagram is monolithic, meaning it consists of one large connected component and some isolated chords. More recently, Acan [1] extended Flajolet and Noy's results about the components of a random chord diagram in several directions, and Acan and Pittel [2] discovered the emergence of a giant component in the intersection graph of a random chord diagram, under some conditions on the number of crossings.…”