Percolation on a product of two trees. (English) Zbl 1243.60078

Kozma, Gady

Ann. Probab. 39, No. 5, 1864-1895 (2011).

Using the standard concepts of percolation theory (see, e.g., [G. Grimmett, Percolation. 2nd ed. Berlin: Springer. (1999; Zbl 0926.60004]) the author studies the triangle condition validity. One says that a transitive graph \(G\) satisfies the triangle condition at some \(p\) if \[ \nabla_p:=\sum_{x,y\in G}\mathbb{P}_p(0\longleftrightarrow x)\mathbb{P}_p(x\longleftrightarrow y) \mathbb{P}_p(0\longleftrightarrow y)<\infty. \] M. Aizenman and C. M. Newman [J. Stat. Phys. 36, 107–143 (1984; Zbl 0586.60096] suggested this condition as a marker for “mean-field behavior”. In particular, they proved that if \(\nabla_{p_c}<\infty\) then \(\mathbb{E}_p|C(0)|\approx (p_c-p)^{-1}\) as \(p\) tends to the critical probability \(p_c\) from below. Here \(C(0)=\{x:0\longleftrightarrow x\}\) and \(|C(0)|\) is its size. In the present paper G. Kozma established that if \(T\) is a regular tree of degree greater than two, then the product graph \(T\times T\) satisfies the triangle condition at \(p_c\). The proof does not examine the degree of vertices and is not “perturbative” in any sense. It relies on an unpublished lemma of O. Schramm.

Reviewer: Alexander V. Bulinski (Moskva)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60B99	Probability theory on algebraic and topological structures

Keywords:

percolation on groups; triangle condition; mean field; product of trees

Citations:

Zbl 0926.60004; Zbl 0586.60096

References:

[1]	Aizenman, M. and Barsky, D. J. (1987). Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 489-526. · Zbl 0618.60098 · doi:10.1007/BF01212322
[2]	Aizenman, M. and Newman, C. M. (1984). Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. 36 107-143. · Zbl 0586.60096 · doi:10.1007/BF01015729
[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. · Zbl 1214.82028 · doi:10.1007/s10955-007-9459-x
[4]	Barsky, D. J. and Aizenman, M. (1991). Percolation critical exponents under the triangle condition. Ann. Probab. 19 1520-1536. · Zbl 0747.60093 · doi:10.1214/aop/1176990221
[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. · Zbl 0961.60015 · doi:10.1214/aop/1022677450
[6]	Grimmett, G. (1999). Percolation , 2nd ed. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 321 . Springer, Berlin. · Zbl 0926.60004
[7]	Hara, T. and Slade, G. (1990). Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128 333-391. · Zbl 0698.60100 · doi:10.1007/BF02108785
[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. · Zbl 1152.82007 · doi:10.1007/s10955-008-9580-5
[9]	Kozma, G. Percolation on a product of two trees, 1st version. Available at . · Zbl 1243.60078
[10]	Kozma, G. (2011). The triangle and the open triangle. Ann. Inst. H. Poincaré Probab. Statist. To appear. Available at . · Zbl 1221.60140
[11]	Kozma, G. and Nachmias, A. (2009). The Alexander-Orbach conjecture holds in high dimensions. Invent. Math. 178 635-654. · Zbl 1180.82094 · doi:10.1007/s00222-009-0208-4
[12]	Kozma, G. and Nachmias, A. (2011). Arm exponents in high dimensional percolation. J. Amer. Math. Soc. 24 375-409. · Zbl 1219.60085 · doi:10.1090/S0894-0347-2010-00684-4
[13]	Lyons, R. and Peres, Y. (2011). Probability on Trees and Networks . Cambridge Univ. Press, Cambridge. To appear. Current version available at . · Zbl 1376.05002
[14]	Nguyen, B. G. (1987). Gap exponents for percolation processes with triangle condition. J. Stat. Phys. 49 235-243. · Zbl 0962.82521 · doi:10.1007/BF01009960
[15]	Schonmann, R. H. (2001). Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Comm. Math. Phys. 219 271-322. · Zbl 1038.82037 · doi:10.1007/s002200100417
[16]	Schonmann, R. H. (2002). Mean-field criticality for percolation on planar non-amenable graphs. Comm. Math. Phys. 225 453-463. · Zbl 0990.82027 · doi:10.1007/s002200100587
[17]	Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138 . Cambridge Univ. Press, Cambridge. · Zbl 0951.60002 · doi:10.1017/CBO9780511470967
[18]	Wu, C. C. (1993). Critical behavior or percolation and Markov fields on branching planes. J. Appl. Probab. 30 538-547. · Zbl 0787.60124 · doi:10.2307/3214764