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Long-range properties of spanning trees. (English) Zbl 0977.82011

Summary: We compute some large-scale properties of the uniform spanning tree process on bounded regions in \(\mathbb{Z}^2\). In particular, we compute the distribution of the meeting point of the branches of the tree issued from three boundary points. We also compute the crossing probabilities of branches of the tree on rectangular and annular regions, as well as the winding number of the branches of the tree.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B05 Classical equilibrium statistical mechanics (general)
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