Local Structure of Sprinkled Causal Sets

TL;DR

This paper investigates the structure of sprinkled causal sets in various spacetimes through numerical and analytical methods, focusing on the properties of the past sets and their implications for quantum gravity models.

Contribution

It introduces criteria for selecting preferred pasts in causal sets and analyzes their properties, advancing the understanding of causal set structure and its applications in spacetime modeling.

Findings

A criterion effectively identifies preferred pasts with desirable properties.

Computed probabilities for specific causal set isomorphism classes.

Estimated the size of past infinity relative to the entire causal set.

Abstract

We describe numerical and analytical investigations of causal sets sprinkled into spacetime manifolds. The first part of the paper is a numerical study of finite causal sets sprinkled into Alexandrov subsets of Minkowski spacetime of dimensions $1 + 1$, $1 + 2$ and $1 + 3$. In particular we consider the rank 2 past of sprinkled causet events, which is the set of events that are two links to the past. Assigning one of the rank 2 past events as `preferred past' for each event yields a `preferred past structure', which was recently proposed as the basis for a causal set d'Alembertian. We test six criteria for selecting rank 2 past subsets. One criterion performs particularly well at uniquely selecting -- with very high probability -- a preferred past satisfying desirable properties. The second part of the paper concerns (infinite) sprinkled causal sets for general spacetime manifolds. After reviewing the construction of the sprinkling process with the Poisson measure, we consider various specific applications. Among other things, we compute the probability of obtaining a sprinkled causal set of a given isomorphism class by combinatorial means, using a correspondence between causal sets in Alexandrov subsets of $1 + 1$ dimensional Minkowski spacetime and 2D-orders. These methods are also used to compute the expected size of the past infinity as a proportion of the total size of a sprinkled causal set.