Path length distribution in two-dimensional causal sets

TL;DR

This paper analyzes the distribution of maximal chain lengths in 2D causal sets to assess their embeddability in Minkowski space and determine their dimensionality, providing theoretical and simulated insights.

Contribution

It derives a recursion relation for expected maximal chain counts in 2D causal sets and links distribution features to embeddability and dimensionality assessment.

Findings

Identifies peak path length and width as measures for embeddability and dimension.

Derives a recursion relation for expected chain counts in 2D causal sets.

Demonstrates differences in distributions for manifoldlike and non-manifoldlike causal sets.

Abstract

We study the distribution of maximal-chain lengths between two elements of a causal set and its relationship with the embeddability of the causal set in a region of flat spacetime. We start with causal sets obtained from uniformly distributed points in Minkowski space. After some general considerations we focus on the 2-dimensional case and derive a recursion relation for the expected number of maximal chains $n_k$ as a function of their length $k$ and the total number of points $N$ between the maximal and minimal elements. By studying these theoretical distributions as well as ones generated from simulated sprinklings in Minkowski space we identify two features, the most probable path length or peak of the distribution $k_0$ and its width $\Delta$, which can be used both to provide a measure of the embeddability of the causal set as a uniform distribution of points in Minkowski space and to determine its dimensionality, if the causal set is manifoldlike in that sense. We end with a few simple examples of $n_k$ distributions for non-manifoldlike causal sets.