On the causal set–continuum correspondence

Abstract

We present two results which concern certain aspects of the question: when is\na causal set well approximated by a Lorentzian manifold? The first result is a\ntheorem which shows that the number-volume correspondence, if required to hold\neven for arbitrarily small regions, is best realized via Poisson sprinkling.\nThe second result concerns a family of lattices in $1+1$ dimensional Minkowski\nspace, known as Lorentzian lattices, which we show provide a much better\nnumber-volume correspondence than Poisson sprinkling for large volumes. We\nargue, however, that this feature should not persist in higher dimensions. We\nconclude by conjecturing a form of the aforementioned theorem that holds under\nweaker assumptions, namely that Poisson sprinkling provides the best\nnumber-volume correspondence in $3+1$ dimensions for spacetime regions with\nmacroscopically large volumes.\n