Continuum limits for classical sequential growth models

• Authors: Graham Brightwell;Nicholas Georgiou
• Venue: Random Structures & Algorithms
• Year: 2010

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. Rideout and Sorkin [Phys. Rev. D 63 (2001) 104011] provide computational evidence that suitably normalized 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 spacetime. Our results indicate that this class of models does not contain any that are good approximations to Minkowski space in any dimension ≥ 2. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010