Minkowski space is locally the Noldus limit of a Poisson process causet

TL;DR

This paper proves that a Poisson process on Minkowski space, when restricted to a compact set, converges in probability to the Minkowski metric using the Noldus limit, linking discrete causal structures to continuous spacetime geometry.

Contribution

It establishes a rigorous convergence result showing that Poisson process causets approximate Minkowski space in the Noldus limit, connecting discrete causal sets to continuous spacetime geometry.

Findings

Poisson process causets converge to Minkowski space in the Noldus limit

Discrete causal distances approximate Minkowski metric on compact sets

Provides a probabilistic framework for causal set continuum limit

Abstract

A poisson process $P_{\lambda}$ on $\mathbb{R}^{d}$ with causal structure inherited from the the usual Minkowski metric on $\mathbb{R}^{d}$ has a normalised discrete causal distance $D_{\lambda}(x,y)$ given by the height of the longest causal chain normalised by $\lambda^{1/d}c_{d}$. We prove that $P_{\lambda}$ restricted to a compact set $Q$ converges in probability in the sense of Noldus to $Q$ with the Minkowksi metric.