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

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

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

DOI: 10.1214/09-AIHP352
Classification: 60K35, 82B43, 20P05, 47N30
Keywords: 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},
     mrnumber = {2779397},
     zbl = {1221.60140},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/09-AIHP352/}
}
TY  - JOUR
AU  - Kozma, Gady
TI  - The triangle and the open triangle
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2011
SP  - 75
EP  - 79
VL  - 47
IS  - 1
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/09-AIHP352/
DO  - 10.1214/09-AIHP352
LA  - en
ID  - AIHPB_2011__47_1_75_0
ER  - 
%0 Journal Article
%A Kozma, Gady
%T The triangle and the open triangle
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2011
%P 75-79
%V 47
%N 1
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/09-AIHP352/
%R 10.1214/09-AIHP352
%G en
%F AIHPB_2011__47_1_75_0
Kozma, Gady. The triangle and the open triangle. Annales de l'I.H.P. Probabilités et statistiques, Volume 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 | Zbl

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

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

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

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

[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 | Zbl

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

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

[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 | Zbl

[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

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

[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 | Zbl

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

Cited by Sources: