×

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

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 It relies on an unpublished lemma of O. Schramm.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60B99 Probability theory on algebraic and topological structures
PDF BibTeX XML Cite
Full Text: DOI arXiv

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
[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
[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
[4] Barsky, D. J. and Aizenman, M. (1991). Percolation critical exponents under the triangle condition. Ann. Probab. 19 1520-1536. · Zbl 0747.60093
[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
[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
[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
[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
[12] Kozma, G. and Nachmias, A. (2011). Arm exponents in high dimensional percolation. J. Amer. Math. Soc. 24 375-409. · Zbl 1219.60085
[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
[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
[16] Schonmann, R. H. (2002). Mean-field criticality for percolation on planar non-amenable graphs. Comm. Math. Phys. 225 453-463. · Zbl 0990.82027
[17] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138 . Cambridge Univ. Press, Cambridge. · Zbl 0951.60002
[18] Wu, C. C. (1993). Critical behavior or percolation and Markov fields on branching planes. J. Appl. Probab. 30 538-547. · Zbl 0787.60124
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.