The triangle and the open triangle
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 75-79.

Nous montrons que dans le cas de la percolation sur un graphe transitif la “condition du triangle” est équivalente à celle du “triangle ouvert”.

We show that for percolation on any transitive graph, the triangle condition implies the open triangle condition.

DOI : https://doi.org/10.1214/09-AIHP352
Classification : 60K35,  82B43,  20P05,  47N30
Mots clés : percolation, Cayley graph, mean-field, triangle condition, operator theory, spectral theory
@article{AIHPB_2011__47_1_75_0,
     author = {Kozma, Gady},
     title = {The triangle and the open triangle},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {75--79},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {1},
     year = {2011},
     doi = {10.1214/09-AIHP352},
     zbl = {1221.60140},
     mrnumber = {2779397},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/09-AIHP352/}
}
Kozma, Gady. The triangle and the open triangle. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 75-79. doi : 10.1214/09-AIHP352. http://archive.numdam.org/articles/10.1214/09-AIHP352/

[1] M. Aizenman and C. M. Newman. Tree graph inequalities and critical behavior in percolation models. J. Statist. Phys. 36 (1984) 107-143. | MR 762034 | Zbl 0586.60096

[2] D. J. Barsky and M. Aizenman. Percolation critical exponents under the triangle condition. Ann. Probab. 19 (1991) 1520-1536. | MR 1127713 | Zbl 0747.60093

[3] B. Bollobás and O. Riordan. Percolation. Cambridge Univ. Press, New York, 2006. | MR 2283880 | Zbl 1118.60001

[4] Y. Eidelman, V. Milman and A. Tsolomitis. Functional Analysis. An Introduction. Graduate Studies in Mathematics 66. Amer. Math. Soc., Providence, RI, 2004. | MR 2067694 | Zbl 1077.46001

[5] G. Grimmett. Percolation, 2nd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer-Verlag, Berlin, 1999. | MR 1707339

[6] T. Hara, R. Van Der Hofstad and G. Slade. Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31 (2003) 349-408. | MR 1959796 | Zbl 1044.82006

[7] T. Hara and G. Slade. Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128 (1990) 333-391. | MR 1043524 | Zbl 0698.60100

[8] W. Hebisch and L. Saloff-Coste. Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 (1993) 673-709. | MR 1217561 | Zbl 0776.60086

[9] M. Heydenreich, R. Van Der Hofstad and A. Sakai. Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk. J. Statist. Phys. 132 (2008) 1001-1049. | MR 2430773 | Zbl 1152.82007

[10] G. Kozma. Percolation on a product of two trees. In preparation.

[11] G. Kozma and A. Nachmias. The Alexander-Orbach conjecture holds in high dimensions. Invent. Math. 178 (2009) 635-654. | Zbl 1180.82094

[12] B. G. Nguyen. Gap exponents for percolation processes with triangle condition. J. Statist. Phys. 49 (1987) 235-243. | MR 923855 | Zbl 0962.82521

[13] R. H. Schonmann. Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Comm. Math. Phys. 219 (2001) 271-322. | MR 1833805 | Zbl 1038.82037

[14] R. H. Schonmann. Mean-field criticality for percolation on planar non-amenable graphs. Comm. Math. Phys. 225 (2002) 453-463. | MR 1888869 | Zbl 0990.82027