Statistics on Linear Chord Diagrams

Linear chord diagrams are partitions of $\left[2n\right]$ into $n$ blocks of size two called chords. We refer to a block of the form $\{i,i+1\}$ as a short chord. In this paper, we study the distribution of the number of short chords on the set of linear chord diagrams, as a generalization of the Narayana distribution obtained when restricted to the set of noncrossing linear chord diagrams. We provide a combinatorial proof that this distribution is unimodal and has an expected value of one. We also study the number of pairs $(i,i+1)$ where $i$ is the minimal element of a chord and $i+1$ is the maximal element of a chord. We show that the distribution of this statistic on linear chord diagrams corresponds to the second-order Eulerian triangle and is log-concave.Comment: 10 pages, final revision

Terminal chords in connected chord diagrams

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.Comment: 25 page

A Correspondence between Chord Diagrams and Families of 0-1 Young Diagrams

A chord diagram is a set of chords in which no pair of chords has a common endvertex. For a chord diagram $E$ having a crossing $S = \{ ac, bd \}$, by the chord expansion of $E$ with respect to $S$, we have two chord diagrams $E_1 = (E\setminus S) \cup \{ ab, cd \}$ and $E_2 = (E\setminus S) \cup \{ da, bc \}$. Starting from a chord diagram $E$, by iterating expansions, we have a binary tree $T$ such that $E$ is a root of $T$ and a multiset of nonintersecting chord diagrams appear in the set of leaves of $T$. The number of leaves, which is not depending on the choice of expansions, is called the chord expansion number of $E$. A $0$-$1$ Young diagram is a Young diagram having a value of $0$ or $1$ for all boxes. This paper shows that the chord expansion number of some type counts the number of $0$-$1$ Young diagrams under some conditions. In particular, it is shown that the chord expansion number of an $n$-crossing, which corresponds to the Euler number, equals the number of $0$-$1$ Young diagrams of shape $(n,n-1,\ldots,1)$ such that each column has at most one $1$ and each row has an even number of $1$'s.Comment: 11 pages, 10 figure

Genus Ranges of Chord Diagrams

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here, where bands are possibly attached to the outer boundary circle of the annulus. The genus range of a chord diagram is the genus values over all such variations of surfaces thus obtained from a given chord diagram. Genus ranges of chord diagrams for a fixed number of chords are studied. Integer intervals that can, and cannot, be realized as genus ranges are investigated. Computer calculations are presented, and play a key role in discovering and proving the properties of genus ranges.Comment: 12 pages, 8 figure

The bulk Hilbert space of double scaled SYK

The emergence of the bulk Hilbert space is a mysterious concept in holography. In arXiv:1811.02584, the SYK model was solved in the double scaling limit by summing chord diagrams. Here, we explicitly construct the bulk Hilbert space of double scaled SYK by slicing open these chord diagrams; this Hilbert space resembles that of a lattice field theory where the length of the lattice is dynamical and determined by the chord number. Under a calculable bulk-to-boundary map, states of fixed chord number map to particular entangled 2-sided states with a corresponding size. This bulk reconstruction is well-defined even when quantum gravity effects are important. Acting on the double scaled Hilbert space is a Type II$_1$ algebra of observables, which includes the Hamiltonian and matter operators. In the appropriate quantum Schwarzian limit, we also identify the JT gravitational algebra including the physical SL(2,R) symmetry generators, and obtain explicit representations of the algebra using chord diagram techniques.Comment: 31 pages, 12 figures; v2-v4: fewer typos, more refs and clarification