Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams

TL;DR

This paper studies the distribution of bridges in linear chord diagrams with bubbles, introduces crystallized diagrams, and provides asymptotic normality results for the number of bridges and short chords.

Contribution

It extends previous work by analyzing bridge distribution in large diagrams and introduces crystallized diagrams with enumeration and distribution results.

Findings

Number of bridges is asymptotically normal with specific mean and variance.

Number of short chords in large diagrams is normally distributed, peaking at √(2n/log n).

Provides formulas for mean and variance related to bubble size and diagram parameters.

Abstract

In a linear chord diagram a short chord joins adjacent vertices while a bubble is a region devoid of short chords. We define a bridge to be a chord joining a vertex interior to a bubble to one exterior to it. Building on earlier work, we investigate the distribution of bridges in the limit of large bubbles and diagrams, and show that the number of bridges is asymptotically normal, obtaining expressions for the associated mean and variance as a function of bubble size. We introduce the notion of a crystallized diagram, defined by the criteria that all its chords are either short or are bridges. We count the number of crystallized diagrams by the number of short chords they contain, and provide the asymptotic distribution in the limit of large crystallized diagrams. We show that for very large diagrams, the number of short chords is normal, and sharply peaked at $\sqrt{2n/\log n}$, where $n$ is the total number of chords in the diagram.