The Annals of Probability

Percolation on a product of two trees

Gady Kozma

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We show that critical percolation on a product of two regular trees of degree ≥ 3 satisfies the triangle condition. The proof does not examine the degrees of vertices and is not “perturbative” in any sense. It relies on an unpublished lemma of Oded Schramm.

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Ann. Probab. Volume 39, Number 5 (2011), 1864-1895.

First available in Project Euclid: 18 October 2011

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60B99: None of the above, but in this section

Percolation on groups triangle condition nonamenable groups mean-field product of trees


Kozma, Gady. Percolation on a product of two trees. Ann. Probab. 39 (2011), no. 5, 1864--1895. doi:10.1214/10-AOP618.

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  • [1] Aizenman, M. and Barsky, D. J. (1987). Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 489–526.
  • [2] Aizenman, M. and Newman, C. M. (1984). Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. 36 107–143.
  • [3] Antunović, T. and Veselić, I. (2008). Sharpness of the phase transition and exponential decay of the subcritical cluster size for percolation and quasi-transitive graphs. J. Stat. Phys. 130 983–1009.
  • [4] Barsky, D. J. and Aizenman, M. (1991). Percolation critical exponents under the triangle condition. Ann. Probab. 19 1520–1536.
  • [5] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. 27 1347–1356.
  • [6] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
  • [7] Hara, T. and Slade, G. (1990). Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128 333–391.
  • [8] Heydenreich, M., van der Hofstad, R. and Sakai, A. (2008). Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk. J. Stat. Phys. 132 1001–1049.
  • [9] Kozma, G. Percolation on a product of two trees, 1st version. Available at
  • [10] Kozma, G. (2011). The triangle and the open triangle. Ann. Inst. H. Poincaré Probab. Statist. To appear. Available at
  • [11] Kozma, G. and Nachmias, A. (2009). The Alexander–Orbach conjecture holds in high dimensions. Invent. Math. 178 635–654.
  • [12] Kozma, G. and Nachmias, A. (2011). Arm exponents in high dimensional percolation. J. Amer. Math. Soc. 24 375–409.
  • [13] Lyons, R. and Peres, Y. (2011). Probability on Trees and Networks. Cambridge Univ. Press, Cambridge. To appear. Current version available at
  • [14] Nguyen, B. G. (1987). Gap exponents for percolation processes with triangle condition. J. Stat. Phys. 49 235–243.
  • [15] Schonmann, R. H. (2001). Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Comm. Math. Phys. 219 271–322.
  • [16] Schonmann, R. H. (2002). Mean-field criticality for percolation on planar non-amenable graphs. Comm. Math. Phys. 225 453–463.
  • [17] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge.
  • [18] Wu, C. C. (1993). Critical behavior or percolation and Markov fields on branching planes. J. Appl. Probab. 30 538–547.