Terminal chords in connected chord diagrams (Q679656)

A chord diagram with \(n\) chords is a perfect matching of the set \(\{1,2,\dots,2n\}\). The oriented intersection graph of a chord diagram \(C\) is the digraph with a vertex for each chord of \(C\) and an oriented edge from chord \(\{a,b\}\) to chord \(\{c,d\}\) whenever \(a<c<b<d\). A chord is terminal if its vertex in the oriented intersection graph has no outgoing edges. Chord diagrams are connected if their oriented intersection graphs are connected. It is known that rooted connected chord diagrams index solutions to certain Dyson-Schwinger equations from quantum field theory. In this paper, the authors study some parameters chracterizing combinatorial properties of rooted connected chord diagrams. Some results are stated in terms of probabilistic limit theorems under the assumption that the underlying diagrams are selected uniformly at random. From the authors' abstract: ``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 the next-to\(^i\) leadinng \(\log\) expansion of the solution to these Dyson-Schwinger equations, and have asymptotic information about the coefficients of the \(\log\) expansions.''