Comptes Rendus
Probability theory
Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters
Comptes Rendus. Mathématique, Volume 354 (2016) no. 9, pp. 944-947.

We prove that critical percolation on any quasi-transitive graph of exponential volume growth does not have a unique infinite cluster. This allows us to deduce from earlier results that critical percolation on any graph in this class does not have any infinite clusters. The result is new when the graph in question is either amenable or nonunimodular.

Nous démontrons que la percolation critique sur les graphes quasi transitifs à croissance exponentielle ne possède pas de composante connexe infinie unique. En utilisant certains résultats antérieurs, ceci nous permet de déduire la non-existence d'une composante connexe infinie pour la percolation critique sur de tels graphes. Ce résultat était auparavant inconnu pour les cas moyennable et non unimodulaire.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.07.013

Tom Hutchcroft 1

1 Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada
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Tom Hutchcroft. Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters. Comptes Rendus. Mathématique, Volume 354 (2016) no. 9, pp. 944-947. doi : 10.1016/j.crma.2016.07.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.07.013/

[1] M. Aizenman; D.J. Barsky Sharpness of the phase transition in percolation models, Commun. Math. Phys., Volume 108 (1987) no. 3, pp. 489-526

[2] T. Antunović; I. Veselić Sharpness of the phase transition and exponential decay of the subcritical cluster size for percolation on quasi-transitive graphs, J. Stat. Phys., Volume 130 (2008) no. 5, pp. 983-1009

[3] L. Bartholdi; B. Virág et al. Amenability via random walks, Duke Math. J., Volume 130 (2005) no. 1, pp. 39-56

[4] I. Benjamini; R. Lyons; Y. Peres; O. Schramm et al. Critical percolation on any nonamenable group has no infinite clusters, Ann. Probab., Volume 27 (1999) no. 3, pp. 1347-1356

[5] I. Benjamini; R. Lyons; O. Schramm Percolation perturbations in potential theory and random walks, Random Walks and Discrete Potential Theory, 1999, pp. 56-84

[6] I. Benjamini; O. Schramm Percolation beyond Zd, many questions and a few answers, Electron. Commun. Probab., Volume 1 (1996) no. 8, pp. 71-82

[7] R.M. Burton; M. Keane Density and uniqueness in percolation, Commun. Math. Phys., Volume 121 (1989) no. 3, pp. 501-505

[8] C. Chou et al. Elementary amenable groups, Ill. J. Math., Volume 24 (1980) no. 3, pp. 396-407

[9] H. Duminil-Copin; V. Tassion A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model, Commun. Math. Phys., Volume 343 (2016) no. 2, pp. 725-745

[10] R. Fitzner; R. van der Hofstad Nearest-neighbor percolation function is continuous for d>10, 2015 (arXiv preprint) | arXiv

[11] A. Gandolfi; M. Keane; C. Newman Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses, Probab. Theory Relat. Fields, Volume 92 (1992) no. 4, pp. 511-527

[12] G.R. Grimmett Percolation, Grundlehren Math. Wiss., 2010

[13] T. Hara; G. Slade Mean-field behaviour and the lace expansion, Probability and Phase Transition, Springer, 1994, pp. 87-122

[14] T.E. Harris A Lower Bound for the Critical Probability in a Certain Percolation Process, Proc. Camb. Philos. Soc., vol. 56, Cambridge Univ. Press, 1960 (p. 3)

[15] K. Juschenko; N. Monod Cantor systems, piecewise translations and simple amenable groups, 2012 (arXiv preprint) | arXiv

[16] R. Lyons Random walks and the growth of groups, C. R. Acad Sci., Ser. I, Volume 320 (1995) no. 11, pp. 1361-1366

[17] R. Lyons; Y. Peres Poisson boundaries of lamplighter groups: proof of the Kaimanovich–Vershik conjecture, 2015 (arXiv preprint) | arXiv

[18] R. Lyons; Y. Peres Probability on Trees and Networks, Cambridge University Press, 2016 http://pages.iu.edu/~rdlyons/ (available at)

[19] R. Lyons; Y. Peres; O. Schramm Minimal spanning forests, Ann. Probab., Volume 34 (2006) no. 5, pp. 1665-1692

[20] J. Milnor et al. Growth of finitely generated solvable groups, J. Differ. Geom., Volume 2 (1968) no. 4, pp. 447-449

[21] C. Newman; L. Schulman Infinite clusters in percolation models, J. Stat. Phys., Volume 26 (1981) no. 3, pp. 613-628

[22] Y. Peres; G. Pete; A. Scolnicov Critical percolation on certain nonunimodular graphs, N.Y. J. Math., Volume 12 (2006), pp. 1-18

[23] L. Russo On the critical percolation probabilities, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 56 (1981) no. 2, pp. 229-237

[24] Á. Timár Percolation on nonunimodular transitive graphs, Ann. Probab., Volume 34 (2006) no. 6, pp. 2344-2364

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