We study the continuum limit of the Benincasa-Dowker-Glaser causal set action on a causally convex compact region. In particular, we compute the action of a causal set randomly sprinkled on a small causal diamond in the presence of arbitrary curvature in various spacetime dimensions. In the continuum limit, we show that the action admits a finite limit. More importantly, the limit is composed by an Einstein-Hilbert bulk term as predicted by the Benincasa-Dowker-Glaser action, and a boundary term exactly proportional to the codimension-two joint volume. Our calculation provides strong evidence in support of the conjecture that the Benincasa-Dowker-Glaser action naturally includes codimension-two boundary terms when evaluated on causally convex regions.

We discuss how to define a kinematical horizon entropy on a causal set. We extend a recent definition of horizon molecules to a setting with a null hypersurface crossing the horizon. We argue that, as opposed to the spacelike case, this extension fails to yield an entropy local to the hypersurface-horizon intersection in the continuum limit when the causal set approximates a curved spacetime. We then investigate the entropy defined via the spacetime mutual information between two regions of a causal diamond truncated by a causal horizon, and find it does limit to the area of the intersection.