Comptes Rendus
no. 11
Dynamical systems/Probability theory
Approximations of standard equivalence relations and Bernoulli percolation at pu
[Approximations de relations d'équivalence standard et percolation de Bernoulli à pu]

Présenté par : Étienne Ghys

Damien Gaboriau1 ; Robin Tucker-Drob2
1 CNRS, Unité de mathématiques pures et appliquées, ENS-Lyon, Université de Lyon, France
2 Department of Mathematics, Texas A&M University, College Station, TX, USA
Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1114-1118.

Le but de cette note est d'annoncer certains résultats d'équivalence orbitale, concernant notamment la notion d'approximation de relations d'équivalence standard préservant la mesure de probabilité par suites croissantes de sous-relations, avec application au comportement en pu de la percolation de Bernoulli sur les graphes de Cayley.

The goal of this note is to announce certain results in orbit equivalence theory, especially concerning the approximation of p.m.p. standard equivalence relations by increasing sequences of sub-relations, with application to the behavior of the Bernoulli percolation on Cayley graphs at the threshold pu.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.09.011
Damien Gaboriau 1 ; Robin Tucker-Drob 2

1 CNRS, Unité de mathématiques pures et appliquées, ENS-Lyon, Université de Lyon, France
2 Department of Mathematics, Texas A&M University, College Station, TX, USA 
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     journal = {Comptes Rendus. Math\'ematique},
     pages = {1114--1118},
     publisher = {Elsevier},
     volume = {354},
     number = {11},
     year = {2016},
     doi = {10.1016/j.crma.2016.09.011},
     language = {en},
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Damien Gaboriau; Robin Tucker-Drob. Approximations of standard equivalence relations and Bernoulli percolation at pu. Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1114-1118. doi : 10.1016/j.crma.2016.09.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.09.011/

[1] J. Feldman; C. Moore Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc., Volume 234 (1977) no. 2, pp. 289-324

[2] D. Gaboriau Invariants L2 de relations d'équivalence et de groupes, Publ. Math. Inst. Hautes Études Sci., Volume 95 (2002), pp. 93-150

[3] D. Gaboriau Invariant percolation and harmonic Dirichlet functions, Geom. Funct. Anal., Volume 15 (2005) no. 5, pp. 1004-1051

[4] D. Gaboriau, R. Tucker-Drob, Approximations and dimensions of standard equivalence relations, in preparation.

[5] O. Häggström; Y. Peres Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously, Probab. Theory Relat. Fields, Volume 113 (1999) no. 2, pp. 273-285

[6] A. Ioana; A.S. Kechris; T. Tsankov Sub-equivalence relations and positive-definite functions, Groups Geom. Dyn., Volume 3 (2009) no. 4, pp. 579-625

[7] V.F.R. Jones; K. Schmidt Asymptotically invariant sequences and approximate finiteness, Amer. J. Math., Volume 109 (1987) no. 1, pp. 91-114

[8] R. Lyons; Y. Peres Probability on Trees and Networks, Cambridge University Press, New York, 2017 (pp. xvi + 699)

[9] R. Lyons; O. Schramm Indistinguishability of percolation clusters, Ann. Probab., Volume 27 (1999) no. 4, pp. 1809-1836

[10] D. Ornstein; B. Weiss Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.), Volume 2 (1980) no. 1, pp. 161-164

[11] Y. Peres Percolation on nonamenable products at the uniqueness threshold, Ann. Inst. Henri Poincaré Probab. Stat., Volume 36 (2000) no. 3, pp. 395-406

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