Terminal chords in connected chord diagrams

TL;DR

This paper studies special chords called terminal chords in rooted connected chord diagrams, revealing their statistical properties and implications for solutions to Dyson-Schwinger equations in quantum field theory.

Contribution

It introduces new combinatorial parameters related to terminal chords and analyzes their distributions, providing insights into quantum field theory applications.

Findings

Number of terminal chords is asymptotically Gaussian with logarithmic mean

Average index of first terminal chord is 2n/3

Method to determine next-to-leading log expansions of Dyson-Schwinger solutions

Abstract

Rooted connected chord diagrams form a nice class of combinatorial objects. Recently they were shown to index solutions to certain Dyson-Schwinger equations in quantum field theory. Key to this indexing role are certain special chords which are called terminal chords. Terminal chords provide a number of combinatorially interesting parameters on rooted connected chord diagrams which have not been studied previously. Understanding these parameters better has implications for quantum field theory. Specifically, we show that the distributions of the number of terminal chords and the number of adjacent terminal chords are asymptotically Gaussian with logarithmic means, and we prove that the average index of the first terminal chord is $2n/3$. Furthermore, we obtain a method to determine any next-to${}^i$ leading log expansion of the solution to these Dyson-Schwinger equations, and have asymptotic information about the coefficients of the log expansions.