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 : 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},
     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, 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 | 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

Cité par Sources :