Infinite collision property for the three-dimensional uniform spanning tree. (English) Zbl 1539.60083
Summary: Let \(\mathcal{U}\) be the three-dimensional uniform spanning tree, whose probability law is denoted by \(\mathbf{P}\). For \(\mathbf{P}\)-a.s. realization of \(\mathcal{U}\), the recurrence of the simple random walk on \(\mathcal{U}\) is proved in [I. Benjamini et al., Ann. Probab. 29, No. 1, 1–65 (2001; Zbl 1016.60009)] and it is also demonstrated in
[T. Hutchcroft and Y. Peres, Electron. Commun. Probab. 20, Paper No. 63, 6 p. (2015; Zbl 1329.60357)]
that two independent simple random walks on \(\mathcal{U}\) collide infinitely often. In this paper, we will give a quantitative estimate on the number of collisions of two independent simple random walks on \(\mathcal{U}\), which provides another proof of the infinite collision property of \(\mathcal{U}\).
MSC:
| 60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |
| 05C80 | Random graphs (graph-theoretic aspects) |
| 60K37 | Processes in random environments |
| 82B27 | Critical phenomena in equilibrium statistical mechanics |
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
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