Combinatorial interpretation of the coefficients of the causal set d'Alembertian

TL;DR

The paper addresses the coefficients $C_i^{(d)}$ in the causal-set d'Alembertian $B^{(d)}$, which are given by alternating gamma-factor expressions. It develops a two-phase combinatorial framework based on colored noncrossing partial chord diagrams to realize these coefficients as explicit counts, first connecting the gamma factors to diagram counts via a generating function $B(x,y)$ (phase 1) and then proving that a signed count of refined objects equals $2^{2\lfloor d/2\rfloor+2} C_i^{(d)}$ (phase 2). In even dimensions, the gamma-ratio simplifies to counts of binary strings, leading to a direct binary-string and lattice-walk interpretation that unifies the treatment across parity. The results illuminate the cancellations in the BDG-like expressions and offer a combinatorial lens for the causal-set d'Alembertian, with potential extensions toward decorated intervals or intrinsic causal-set quantities that reproduce the operator without relying on alternating coefficients.

Abstract

The causal set theory d'Alembertian has rational coefficients for which alternating expressions are known. Here, a combinatorial interpretation of these numbers is given.

Figures (4)

• Figure 1: A partial rooted chord diagram. Here there are three chords and nine points. Point 1 is marked with an $\times$ and the rest are labelled counterclockwise, so the chords, written as sets of their endpoints as in the definition are $\{1,3\}$, $\{4,7\}$, and $\{5,6\}$. Points $2$, $8$ and $9$ are bare. This diagram is noncrossing.
• Figure 2: An example of an element of $\mathcal{B}_{5, 12}$. The red and blue chords are also indicated with a heavier weight, and the first ends are circled. This is an element of $\mathcal{B}_{5,12}$ because there are 5 chords total, 12 points total, the inside of the blue chord is entirely filled with black chords and in the trivial sense the same is true of the red chords since they have no points on their inside, and there are no other black chords.
• Figure 3: An example of an element of $\mathcal{B}_{8, 30, 2, 3}$. The insertion places are marked by arrows and the width of each red or blue chords is the number of arrows immediately before its first end. To begin with, we see that this diagram is an element of $\mathcal{B}_{8,30}$ since it has 30 points, 8 chords and satisfies the required conditions. Saying further that it is in $\mathcal{B}_{8,30, 2, 3}$ means that the red width 1 chord and the blue chord both have fewer than $2$ free points immediately before them. It is these two chords that we need to consider because the first three insertion places come before these chords, or equivalently because $w(e_1) = 1 < 3 \leq 4 = w(e_1)+w(e_2)$.
• Figure 4: An example of the sequence of indecomposable pieces making up a block of $1$s in $s(b)$. The arrows indicate the points which directly contribute bits to $s(b)$. Note that each chord has one end which contributes a bit to $s(b)$. Furthermore that the sequence of indecomposable pieces has exactly one internal bare point (which necessarily is not contributing a bit) separating a sequence of indecomposable pieces where the contributing ends are in the even positions from a sequence of indecomposable pieces where the contributing ends are in the odd positions. This will always happen, potentially with the sequence on one side of the internal bare point being empty. The portion of $s(b)$ given by the part of the chord diagram shown in the figure is $011111111110$ (with ten $1$s in the block of $1$s).

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Definition 3
• Lemma 4
• proof
• Lemma 5
• proof
• Definition 6
• Theorem 7
• proof