Charting causal set configuration space with graph observables

Abstract

The configuration space of causal sets is vast. It is a critical goal to map out this space. Here, we take a practical step towards this goal. We investigate nine classes of causal sets, most of them not studied before. These include manifoldlike causal sets with inhomogeneous Ricci curvature, both topologically trivial and nontrivial. We also study classes of non-manifoldlike causal sets, including lattices, layered orders as well as Lorentzian quasicrystals. Finally, we study classes of causal sets that are not manifoldlike, but are expected to become manifoldlike under a suitable coarse-graining process. We use this broad range of distinct classes of causal sets as a testbed for observables. Rather than focusing on continuum-geometry inspired observables, such as curvature invariants, which often exhibit large fluctuations and are computationally very expensive, we focus on graph observables, including some observables that constitute subgraph statistics and some that are global. We find that three observables, namely the link degree distribution, the eigenvalues of the graph Laplacian of the symmetrized Hasse diagram and the recently proposed abundance of causal intervals, can distinguish between the distinct classes of causal sets. This is made possible by the small fluctuations that these observables have in most classes.