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 proporti...

• Type: Journal
• Year: 2010
• Where: RSA
• Authors: Graham Brightwell, Nicholas Georgiou
• Keywords: Classical Sequential Growth, Continuum Limits, RSA 2010, Sequential Growth Models