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, subset of 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\ge 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)\in\R^{d-1}\times 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&#39;, at time r&#39;>r, (x&#39;,r&#39;)\in S, where r&#39; is the minimal time after r such that |x-x&#39;|<1. All the walks jumping to x&#39; at time r&#39; coalesce with the one starting at (x&#39;,r&#39;). Calling (x&#39;,r&#39;) = α(x,r), the graph has vertex set S and edges {(s,α(s)), s\in S}. This enables us to shift the origin of S^o = S + δ_0 (the Palm version of S) to another point in such a way that the distribution of S^o does not change (to any point if d = 2 and d = 3; point-stationarity).