Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams

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.