  In order to understand the detailed mechanism by which a fundamental
discreteness can provide a finite entanglement entropy, we consider the
entanglement entropy of two classes of free massless scalar fields on causal
sets that are well approximated by causal diamonds in Minkowski spacetime of
dimensions 2,3 and 4. The first class is defined from discretised versions of
the continuum retarded Green functions, while the second uses the causal set's
retarded nonlocal d'Alembertians parametrised by a length scale $l_k$. In both
cases we provide numerical evidence that the area law is recovered when the
double-cutoff prescription proposed in arXiv:hep-th/1611.10281 is imposed. We
discuss in detail the need for this double cutoff by studying the effect of two
cutoffs on the quantum field and, in particular, on the entanglement entropy,
in isolation. In so doing, we get a novel interpretation for why these two
cutoff are necessary, and the different roles they play in making the
entanglement entropy on causal sets finite.
