Discrete Spacetime: a Web of Chains

TL;DR

This paper derives improved formulas for chain length distributions in causal sets embedded in Minkowski space, validating them with simulations and exploring their use in identifying manifoldlike structures and estimating dimension.

Contribution

It provides a corrected derivation of chain length distributions for causal sets in Minkowski space and applies these to determine manifoldlikeness and dimensionality.

Findings

Numerical simulations align better with the corrected distributions.

The method helps identify whether a causal set is manifoldlike.

Comparison with other dimension estimation methods shows competitive performance.

Abstract

This paper studies the distribution of chain and maximal chain lengths in a causal set. We first provide a new derivation for these distributions for a causal set uniformly embedded in Minkowski space, for various dimensionalities, which includes a correction to previously available expressions. Results of numerical simulations show a better agreement with the improved theoretical distributions. As examples of applications of those distributions, we then expand on previous work of ours regarding their use in establishing whether a causal set is manifoldlike and, if it is, finding its dimensionality. We then compare this measure of the dimension with other methods in the literature.