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

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.