Annals of Probability

Indistinguishability of the components of random spanning forests

Ádám Timár

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove that the infinite components of the Free Uniform Spanning Forest (FUSF) of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the Free Minimal Spanning Forest (FMSF). We also show that with the above assumptions there can only be 0, 1 or infinitely many components, which solves the problem for the FUSF of Caylay graphs completely. These answer questions by Benjamini, Lyons, Peres and Schramm for Cayley graphs, which have been open up to now. Our methods apply to a more general class of percolations, those satisfying “weak insertion tolerance”, and work beyond Cayley graphs, in the more general setting of unimodular random graphs.

Article information

Ann. Probab., Volume 46, Number 4 (2018), 2221-2242.

Received: January 2017
Revised: August 2017
First available in Project Euclid: 13 June 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 82B43: Percolation [See also 60K35]

Spanning forests uniform spanning forest minimal spanning forest insertion tolerance indistinguishability


Timár, Ádám. Indistinguishability of the components of random spanning forests. Ann. Probab. 46 (2018), no. 4, 2221--2242. doi:10.1214/17-AOP1225.

Export citation


  • [1] Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Probab. 12 1454–1508.
  • [2] Benjamini, I., Kesten, H., Peres, Y. and Schramm, O. (2004). Geometry of the uniform spanning forest: Transitions in dimensions $4,8,12,\dots$. Ann. of Math. (2) 160 465–491.
  • [3] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29–66.
  • [4] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (2001). Uniform spanning forests. Ann. Probab. 29 1–65.
  • [5] Chifan, I. and Ioana, A. (2010). Ergodic subequivalence relations induced by a Bernoulli action. Geom. Funct. Anal. 20 53–67.
  • [6] Gaboriau, D. and Lyons, R. (2009). A measurable-group-theoretic solution to von Neumann’s problem. Invent. Math. 177 533–540.
  • [7] Hutchcroft, T. and Nachmias, A. (2017). Indistinguishability of trees in uniform spanning forests. Probab. Theory Related Fields 168 113–152.
  • [8] Lyons, R. and Peres, Y. (2016). Probability on Trees and Networks. Cambridge Univ. Press, Cambridge.
  • [9] Lyons, R., Peres, Y. and Schramm, O. (2006). Minimal spanning forests. Ann. Probab. 34 1665–1692.
  • [10] Lyons, R. and Schramm, O. (1999). Indistinguishability of percolation clusters. Ann. Probab. 27 1809–1836.
  • [11] Lyons, R. and Schramm, O. (1999). Stationary measures for random walks in a random environment with random scenery. New York J. Math. 5 107–113.
  • [12] Pemantle, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 1559–1574.
  • [13] Timár, Á. (2006). Ends in free minimal spanning forests. Ann. Probab. 34 865–869.
  • [14] Timár, A. (2006). Neighboring clusters in Bernoulli percolation. Ann. Probab. 34 2332–2343.