Poisson trees, succession lines and coalescing random walks. (English) Zbl 1042.60064

Summary: 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 \(S\subset\mathbb{R}^d\). 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 \(d\geq 4\) (a forest); (3) in \(d=2\) and \(d=3\) we construct a bijection between the points of the Poisson process and \(\mathbb{Z}\) using the preorder-traversal algorithm. To construct the graph we interpret each point in \(S\) as a space-time point \((x,r)\in\mathbb{R}^{d-1} \times\mathbb{R}\). 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 \(r'>r\), \((x',r')\in S\), 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 \((x',r')=\alpha (x,r)\), the graph has vertex set \(S\) and edges \(\{(s,\alpha(s)), s\in S\}\). This enables us to shift the origin of \(S^0=S\cup\{0\}\) (the Palm version of \(S)\) to another point in such a way that the distribution of \(S^0\) does not change (to any point if \(d=2\) and \(d=3\); point-stationarity).


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
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