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Ends in free minimal spanning forests. (English) Zbl 1255.60173

Summary: We show that for a transitive unimodular graphs, the number of ends is the same for every tree of the free minimal spanning forest. This answers a question of R. Lyons, Y. Peres and O. Schramm [Ann. Probab. 34, No. 5, 1665–1692 (2006; Zbl 1142.60065)].

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
05A05 Permutations, words, matrices
60D05 Geometric probability and stochastic geometry

Citations:

Zbl 1142.60065

References:

[1] Häggström, O., Peres, Y. and Schonmann, R. (1999). Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness. In Perplexing Probability Problems : Papers in Honor of H. Kesten (M. Bramson and R. Durrett, eds.) 69–90. Birkhäuser, Boston. · Zbl 0948.60098
[2] Lyons, R., Peres, Y. and Schramm, O. (2006). Minimal spanning forests. Ann. Probab. 34 . · Zbl 1142.60065 · doi:10.1214/009117906000000269
[3] Lyons, L. and Schramm, O. (1999). Indistinguishability of percolation clusters. Ann. Probab. 27 1809–1836. · Zbl 0960.60013 · doi:10.1214/aop/1022677549
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