Terminal chords in connected chord diagrams

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.

Specically, 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 coecients of the log expansions.