Poisson trees, succession lines and coalescing random walks

Abstract

We give a deterministic algorithm to construct a graph with no loops (a tree or a forest) whose vertices are the points of a d-dimensional stationary Poisson process S⊂ R d . The algorithm is independent of the origin of coordinates. We show that (1) the graph has one topological end – that is, from any point there is exactly one infinite self-avoiding path; (2) the graph has a unique connected component if d=2 and d=3 (a tree) and it has infinitely many components if d⩾4 (a forest); (3) in d=2 and d=3 we construct a bijection between the points of the Poisson process and Z using the preorder-traversal algorithm. To construct the graph we interpret each point in S as a space-time point (x,r)∈ R d−1× R . Then a ( d−1)-dimensional random walk in continuous time continuous space starts at site x at time r. The first jump of the walk is to point x′, at time r′> r, ( x′, r′)∈ S, where r′ is the minimal time after r such that | x− x′|<1. All the walks jumping to x′ at time r′ coalesce with the one starting at ( x′, r′). Calling ( x′, r′)= α( x, r), the graph has vertex set S and edges {( s, α( s)), s∈ S}. This enables us to shift the origin of S ∘= S∪{0} (the Palm version of S) to another point in such a way that the distribution of S ∘ does not change (to any point if d=2 and d=3; point-stationarity).