ORDER-INVARIANT MEASURES ON FIXED CAUSAL SETS
ORDER-INVARIANT MEASURES ON FIXED CAUSAL SETS
arXiv:0901.0242v1 [math.CO] 2 Jan 2009
GRAHAM BRIGHTWELL AND MALWINA LUCZAK Abstract. A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the ordertype of the natural numbers – we call such a linear extension a natural extension. We study probability measures on the set of natural extensions of a causal set, especially those measures having the property of order-invariance: if we condition on the set of the bottom k elements of the natural extension, each possible ordering among these k elements is equally likely. We give sufficient conditions for the existence and uniqueness of an order-invariant measure on the set of natural extensions of a causal set.
1. Introduction A causal set is a countably infinite partially ordered set P = (Z, λ(j). We call such a linear extension a natural extension. We are interested in probability measures on the set of natural extensions of a fixed causal set P . More specifically, we are interested in such measures that satisfy a property called order-invariance: for each k ∈ N, and each k-element down-set A of P , conditioned on the event that {λ(1), . . . , λ(k)} = A, each possible linear extension of the restriction PA of P to A is equally likely to be the restriction of λ to [k]. This work is part of a wider project, initiated in our companion paper [7]. In that paper, we consider probability measures where the causal set P is also random. More precisely, we consider processes that generate a causal set one element at a time, at each stage adding a maximal element, with a label drawn from a given set (which we take to be [0, 1]), and putting it above some down-set in the current poset. Such processes are called causal set processes: formally they are Markov processes, whose states are pairs (x1 · · · xk ,
arXiv:0901.0242v1 [math.CO] 2 Jan 2009
GRAHAM BRIGHTWELL AND MALWINA LUCZAK Abstract. A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the ordertype of the natural numbers – we call such a linear extension a natural extension. We study probability measures on the set of natural extensions of a causal set, especially those measures having the property of order-invariance: if we condition on the set of the bottom k elements of the natural extension, each possible ordering among these k elements is equally likely. We give sufficient conditions for the existence and uniqueness of an order-invariant measure on the set of natural extensions of a causal set.
1. Introduction A causal set is a countably infinite partially ordered set P = (Z, λ(j). We call such a linear extension a natural extension. We are interested in probability measures on the set of natural extensions of a fixed causal set P . More specifically, we are interested in such measures that satisfy a property called order-invariance: for each k ∈ N, and each k-element down-set A of P , conditioned on the event that {λ(1), . . . , λ(k)} = A, each possible linear extension of the restriction PA of P to A is equally likely to be the restriction of λ to [k]. This work is part of a wider project, initiated in our companion paper [7]. In that paper, we consider probability measures where the causal set P is also random. More precisely, we consider processes that generate a causal set one element at a time, at each stage adding a maximal element, with a label drawn from a given set (which we take to be [0, 1]), and putting it above some down-set in the current poset. Such processes are called causal set processes: formally they are Markov processes, whose states are pairs (x1 · · · xk ,
