Percolation on nonamenable products at the uniqueness threshold
Peres, Yuval
Annales de l'I.H.P. Probabilités et statistiques, Tome 36 (2000), p. 395-406 / Harvested from Numdam
Publié le : 2000-01-01
@article{AIHPB_2000__36_3_395_0,
     author = {Peres, Yuval},
     title = {Percolation on nonamenable products at the uniqueness threshold},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {36},
     year = {2000},
     pages = {395-406},
     mrnumber = {1770624},
     zbl = {0965.60094},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2000__36_3_395_0}
}
Peres, Yuval. Percolation on nonamenable products at the uniqueness threshold. Annales de l'I.H.P. Probabilités et statistiques, Tome 36 (2000) pp. 395-406. http://gdmltest.u-ga.fr/item/AIHPB_2000__36_3_395_0/

[1] Adams S., Lyons R., Amenability, Kazhdan's property and percolation for trees, groups and equivalence relations, Israel J. Math. 75 (1991) 341-370. | MR 1164598 | Zbl 0790.43001

[2] Benjamini I., Lyons R., Peres Y., Schramm O., Group-invariant percolation on graphs, Geom. Funct. Anal. 9 (1999) 29-66. | MR 1675890 | Zbl 0924.43002

[3] Benjamini I., Lyons R., Peres Y., Schramm O., Critical percolation on any nonamenable group has no infinite clusters, Ann. Probab. (1999), to appear. | MR 1733151 | Zbl 0961.60015

[4] Benjamini I., Schramm O., Percolation beyond Zd, many questions and a few answers, Electronic Commun. Probab. 1 (8) (1996) 71-82. | MR 1423907 | Zbl 0890.60091

[5] Burton R.M., Keane M., Density and uniqueness in percolation, Comm. Math. Phys. 121 (1989) 501-505. | MR 990777 | Zbl 0662.60113

[6] Gandolfi A., Keane M.S., Newman C.M., Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses, Probab. Theory Related Fields 92 (1992) 511-527. | MR 1169017 | Zbl 0767.60098

[7] Grimmett G.R., Newman C.M., Percolation in oo + 1 dimensions, in: Grimmett G.R., Welsh D.J.A. (Eds.), Disorder in Physical Systems, Clarendon Press, Oxford, 1990, pp. 167-190. | MR 1064560 | Zbl 0721.60121

[8] Häggström O., Infinite clusters in dependent automorphism invariant percolation on trees, Ann. Probab. 25 (1997) 1423-1436. | MR 1457624 | Zbl 0895.60098

[9] Häggström O., Peres Y., Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously, Probab. Theory Related Fields 113 (1999) 273-285. | MR 1676835 | Zbl 0921.60091

[10] Häggström O., Peres Y., Schonmann R.H., Percolation on transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness, in: Bramson M., Durrett R. (Eds.), Perplexing Probability Problems: Papers in Honor of Harry Kesten, Birkhäuser, 1999, pp. 69-90. | MR 1703125 | Zbl 0948.60098

[11] Lalley S.P., Percolation on Fuchsian groups, Ann. Inst. H. Poincaré Probab. Statist. 34 (1998) 151-177. | Numdam | MR 1614583 | Zbl 0911.60084

[12] Liggett T.M., Multiple transition points for the contact process on the binary tree, Ann. Probab. 24 (1996) 1675-1710. | MR 1415225 | Zbl 0871.60087

[13] Lyons R., Schramm O., Indistinguishability of percolation clusters, Ann. Probab. (1999), to appear. | MR 1742889 | Zbl 0960.60013

[14] Paterson A.L.T., Amenability, American Mathematical Soc., Providence, 1988. | MR 961261 | Zbl 0648.43001

[15] Pemantle R., The contact process on trees, Ann. Probab. 20 (1992) 2089-2116. | MR 1188054 | Zbl 0762.60098

[16] Pemantle R., Peres Y., Nonamenable products are not treeable, Preprint, 1999, Israel J. Math., to appear. | MR 1776080 | Zbl 0961.43002

[17] Salvatori M., On the norms of group-invariant transition operators on graphs, J. Theor. Probab. 5 (1992) 563-576. | MR 1176438 | Zbl 0751.60068

[18] Salzano M., Schonmann R.H., A new proof that for the contact process on homogeneous trees local survival implies complete convergence, Ann. Probab. 26 (1998) 1251-1258. | MR 1640345 | Zbl 0937.60094

[19] Schonmann R.H., Stability of infinite clusters in supercritical percolation, Probab. Theory Related Fields 113 (1999) 287-300. | MR 1676831 | Zbl 0921.60092

[20] Schonmann R.H., Percolation in ∞ + 1 dimensions at the uniqueness threshold, in: Bramson M., Durrett R. (Eds.), Perplexing Probability Problems: Papers in Honor of H. Kesten, Birkhäuser, 1999, pp. 53-67. | MR 1703124 | Zbl 0948.60099

[21] Zhang Y., The complete convergence theorem of the contact process on trees, Ann. Probab. 24 (1996) 1408-1443. | MR 1411500 | Zbl 0876.60092