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).

MSC:

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