Order-invariant measures on causal sets

Abstract

A causal set is a partially ordered set on a countably infinite ground-set such that each element is above finitely many others. A natural extension of a causal set is an enumeration of its elements which respects the order. We bring together two different classes of random processes. In one class, we are given a fixed causal set, and we consider random natural extensions of this causal set: we think of the random enumeration as being generated one point at a time. In the other class of processes, we generate a random causal set, working from the bottom up, adding one new maximal element at each stage. Processes of both types can exhibit a property called order-invariance: if we stop the process after some fixed number of steps, then, conditioned on the structure of the causal set, every possible order of generation of its elements is equally likely. We develop a framework for the study of order-invariance which includes both types of example: order-invariance is then a property of probability measures on a certain space. Our main result is a description of the extremal order-invariant measures.