ec 2 00 3 POISSON TREES , SUCCESSION LINES AND COALESCING RANDOM WALKS

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