Renewal structure and critical dimension of causal set d'Alembertians

Plain-language summary Causal set theory models spacetime as a discrete collection of events ordered by cause and effect. On such a set, the wave operator, the basic rule for how a field propagates, has to be built from counts of events in the past. These constructions are nonlocal: the value at one event depends on events far back along its light cone. Earlier work found that the standard four-dimensional construction allows solutions that grow exponentially in time, while the two-dimensional one does not. This paper rewrites the stability question exactly as a renewal equation, the equation used for example in population dynamics, except that the weights here can be negative. In this form two requirements separate clearly. Reproducing the ordinary wave equation at long wavelengths fixes a few moments of the weights and forces some weights to be negative. Stability is a different condition that depends on the whole weight profile. For the most economical family of constructions the two requirements conflict above a critical dimension, and in the limit of many cancellations only two dimensions stays stable. Other four-dimensional constructions with more layers are stable in our numerical tests, and on randomly generated causal sets with enough smoothing their simulated fields decay over the simulated time, while the standard operator shows its growing mode. The main message is that a correct long-wavelength limit does not by itself guarantee a stable time evolution. Why it matters Causal set theory needs a wave operator to describe how fields propagate, and the same operators enter the definition of its gravitational action. An operator with growing modes cannot play that role, because a small disturbance grows exponentially. The renewal form turns stability into a computable test on the weight profile, places the known instability of the four-dimensional operator in a family with a critical dimension, and shows that stable four-dimensional alternatives exist. More broadly, matching a known equation at long wavelengths is the usual check for a discrete or nonlocal model; the results show that this check is not enough, and that stability has to be verified as a separate condition.