Simple indicators for Lorentzian causets

TL;DR

This paper introduces statistical indicators, including longest/shortest path plots, to compare directed acyclic graphs with ideal Lorentzian causets, helping assess their similarity to spacetime models and detect non-locality.

Contribution

The paper proposes new simple indicators, especially longest/shortest path plots, for evaluating how closely causets resemble Lorentzian spacetime structures.

Findings

Indicators effectively distinguish Lorentzian causets from non-Lorentzian graphs.

Deterministic causets can exhibit optimal Lorentzianity according to the new indicators.

The tools can identify non-locality and regularity in causet structures.

Abstract

Several classes of directed acyclic graphs have been investigated in the last two decades, in the context of the Causal Set Program, in search for good discrete models of spacetime. We introduce some statistical indicators that can be used for comparing these graphs and for assessing their closeness to the ideal Lorentzian causal sets ('causets') -- those obtained by sprinkling points in a Lorentzian manifold. In particular, with the reversed triangular inequality of Special Relativity in mind, we introduce 'longest/shortest path plots', an easily implemented tool to visually detect the extent to which a generic causet matches the wide range of path lengths between events of Lorentzian causets. This tool can attribute some degree of 'Lorentzianity' - in particular 'non-locality' - also to causets that are not (directly) embeddable and that, due to some regularity in their structure, would not pass the key test for Lorentz invariance: the absence of preferred reference frames. We compare the discussed indicators and use them for assessing causets both of stochastic and of deterministic, algorithmic origin, finding examples of the latter that behave optimally w.r.t. our longest/shortest path plots.