Invariant percolation and harmonic Dirichlet functions

Geometric and Functional Analysis

Volumn 15, Issue 5, 2005, Pages 1004-1051

  Gaboriau, D ^a  

^a UNITÉ DE MATHÉMATIQUES PURES ET APPLIQUÉES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 30144438419     PISSN: 1016443X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00039-005-0539-2     Document Type: Article

References (33)

1. [A] S. ADAMS, Trees and amenable equivalence relations, Ergodic Theory Dynamical Systems, 10:1 (1990), 1-14.
2. [At] M. ATIYAH, Elliptic operators, discrete groups and von Neumann algebras, in "Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974)", Astérisque, SMF 32-33 (1976), 43-72.
3. [BB] E. BABSON, I. BENJAMINI, Cut sets and normed cohomology with applications to percolation, Proc. Amer. Math. Soc. 127:2 (1999), 589-597.
4. 2-Betti number, Potential Anal. 6:4 (1997), 313-326.
5. [BenLPS1] I. BENJAMINI, R. LYONS, Y. PERES, O. SCHRAMM, Group-invariant percolation on graphs, GAFA, Geom. funct. anal. 9:1 (1999), 29-66.
6. [BenLPS2] I. BENJAMINI, R. LYONS, Y. PERES, O. SCHRAMM, Critical percolation on any nonamenable group has no infinite clusters, Ann. Probab. 27:3 (1999), 1347-1356.
7. [BenLS] I. BENJAMINI, R. LYONS, O. SCHRAMM, Percolation perturbations in potential theory and random walks, in "Random Walks and Discrete Potential Theory (Cortona, 1997)", Sympos. Math. XXXIX, Cambridge Univ. Press (1999), 56-84.
8. d, many questions and a few answers, Electron. Comm. Probab. 1:8 (1996), 71-82 (electronic).
9. [BenS2] I. BENJAMINI, O. SCHRAMM, Percolation in the hyperbolic plane, Jour. Amer. Math. Soc. 14 (2001), 487-507.
10. [BiBCK] M. BISKUP, C. BORGS, J.T. CHAYES, R. KOTECKÝ, Gibbs states of graphical representations of the Potts model with external fields, J. Math. Phys. 41:3 (2000), 1170-1210.
11. [BuK] R. BURTON, M. KEANE, Density and uniqueness in percolation, Comm. Math. Phys. 121:3 (1989), 501-505.
12. 2-cohomology and group cohomology, Topology 25:2 (1986), 189-215.
13. [FM] J. FELDMAN, C. MOORE, Ergodic equivalence relations, cohomology and von Neumann algebras. I and II, Trans. Amer. Math. Soc., 234:2 (1977), 289-324; 325-359.
14. [FoK] C.M. FORTUIN, P.W. KASTELEYN, On the random-cluster model. I. Introduction and relation to other models, Physica 57 (1972), 536-564.
15. [G1] D. GABORIAU, Coût des relations d'équivalence et des groupes, Invent. Math. 139:1 (2000), 41-98.
16. 2 de relations d'équivalence et de groupes, Publ. Math. Inst. Hautes Étud. Sci. 95:1 (2002), 93-150.
17. [GrN] G.R. GRIMMETT, C.M. NEWMAN, Percolation in ∞ + 1 dimensions, in "Disorder in Physical Systems", Oxford Sci. Publ., Oxford Univ. Press, New York (1990), 167-190.
18. [HJL1] O. HÄGGSTRÖM, J. JONASSON, R. LYONS, Coupling and Bernoullicity in random-cluster and Potts models, Bernoulli 8:3 (2002), 275-294.
19. [HJL2] O. HÄGGSTRÖM, J. JONASSON, R. LYONS, Explicit isoperimetric constants and phase transitions in the random-cluster Model, Ann. of Proba. 30:1 (2002), 443-473.
20. [HP] O. HÄGGSTRÖM, Y. PERES, Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously, Probab. Theory Related Fields 113:2 (1999), 273-285.
21. [HPS] O. HÄGGSTRÖM, Y. PERES, R.H. SCHONMANN, Percolation on transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness, in "Perplexing Problems in Probability", Progr. Probab. 44, Birkhäuser Boston, Boston, MA (1999), 69-90.
22. [K] A. KECHRIS, Classical Descriptive Set Theory, Graduate Texts in Mathematics, Springer-Verlag New York 156 (1995).
23. [L] G. LEVITT, On the cost of generating an equivalence relation, Ergodic Theory Dynam. Systems 15:6 (1995), 1173-1181.
24. [Ly1] R. LYONS, Random walks and the growth of groups, C.R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1361-1366.
25. [Ly2] R. LYONS, Phase transitions on nonamenable graphs, probabilistic techniques in equilibrium and nonequilibrium statistical physics, J. Math. Phys. 41, (2000), 1099-1126.
26. [LyP] R. LYONS, Y. PERES, Probability on Trees and Networks, Cambridge University Press, to appear; http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html.
27. [LyS] R. LYONS, O. SCHRAMM, Indistinguishability of percolation clusters, Ann. Probab. 27:4 (1999), 1809-1836.
28. [M] F. MARTIN, Analyse harmonique et 1-cohomologie réduite des groupes localement compacts, Thèse de l'université de Neuchâtel, 2003.
29. [PaS] I. PAK, T. SMIRNOVA-NAONIBEDA, On non-uniqueness of percolation on nonamenable Cayley graphs, C.R. Acad. Sci. Paris Sér. I Math. 330:6 (2000), 495-500.
30. [Pe] Y. PERES, Percolation on nonamenable products at the uniqueness threshold, Ann. Inst. H. Poincaré Probab. Statist. 36:5 (2000), 395-406.
31. [PeP] Y. PERES, R. PEMANTLE, Nonamenable products are not treeable, Israel J. Math. 118 (2000), 147-155.
32. 2-Betti Numbers of Discrete Measured Groupoids, preprint; arXiv math.AT/0312411.
33. [So] P.M. SOARDI, Rough isometries and Dirichlet finite harmonic functions on graphs, Proc. Amer. Math. Soc. 119:4 (1993), 1239-1248.