Continuum limits for classical sequential growth models

A random graph order, also known as a transitive percolation process, is defined by taking a random graph on the vertex set {0, . . . , n− 1}, and putting i below j if there is a path i = i1 · · · ik = j in the graph with i1 < · · · < ik. In [15], Rideout and Sorkin provide computational evidence that suitably normalised sequences of random graph orders have a “continuum limit”. We confirm that this is the case, and show that the continuum limit is always a semiorder. Transitive percolation processes are a special case of a more general class called classical sequential growth models. We give a number of results describing the large-scale structure of a general classical sequential growth model. We show that for any sufficiently large n, and any classical sequential growth model, there is a semiorder S on {0, . . . , n− 1} such that the random partial order on {0, . . . , n− 1} generated according to the model differs from S on an arbitrarily small proportion of pairs. We also show that, if any sequence of classical sequential growth models has a continuum limit, then this limit is (essentially) a semiorder. We give some examples of continuum limits that can occur. Classical sequential growth models were introduced as the only models satisfying certain properties making them suitable as discrete models for space-time. Our results indicate that this class of models does not contain any that are good approximations to Minkowski space in any dimension ≥ 2.