The Annals of Probability

Locality of percolation for Abelian Cayley graphs

Sébastien Martineau and Vincent Tassion

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove that the value of the critical probability for percolation on an Abelian Cayley graph is determined by its local structure. This is a partial positive answer to a conjecture of Schramm: the function $\mathrm{p}_{\mathrm{c}}$ defined on the set of Cayley graphs of Abelian groups of rank at least $2$ is continuous for the Benjamini–Schramm topology. The proof involves group-theoretic tools and a new block argument.

Article information

Source
Ann. Probab., Volume 45, Number 2 (2017), 1247-1277.

Dates
Received: January 2014
Revised: March 2015
First available in Project Euclid: 31 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1490947319

Digital Object Identifier
doi:10.1214/15-AOP1086

Mathematical Reviews number (MathSciNet)
MR3630298

Zentralblatt MATH identifier
06797091

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65]

Keywords
Percolation Abelian groups graph limits locality

Citation

Martineau, Sébastien; Tassion, Vincent. Locality of percolation for Abelian Cayley graphs. Ann. Probab. 45 (2017), no. 2, 1247--1277. doi:10.1214/15-AOP1086. https://projecteuclid.org/euclid.aop/1490947319


Export citation

References

  • [1] Benjamini, I. (2013). Euclidean vs. graph metric. In Erdös Centennial. Bolyai Soc. Math. Stud. 25 35–57. János Bolyai Math. Soc., Budapest.
  • [2] Benjamini, I., Nachmias, A. and Peres, Y. (2011). Is the critical percolation probability local? Probab. Theory Related Fields 149 261–269.
  • [3] Benjamini, I. and Schramm, O. (1996). Percolation beyond $\mathbf{Z}^{d}$, many questions and a few answers. Electron. Commun. Probab. 1 71–82 (electronic).
  • [4] Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 13 pp. (electronic).
  • [5] Bodineau, T. (2005). Slab percolation for the Ising model. Probab. Theory Related Fields 132 83–118.
  • [6] Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501–505.
  • [7] de Lima, B. N. B., Sanchis, R. and Silva, R. W. C. (2011). Critical point and percolation probability in a long range site percolation model on $\mathbb{Z}^{d}$. Stochastic Process. Appl. 121 2043–2048.
  • [8] Grigorchuk, R. I. (1984). Degrees of growth of finitely generated groups and the theory of invariant means. Math. USSR, Izv. 25 259–300.
  • [9] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften 321. Springer, Berlin.
  • [10] Grimmett, G. R. and Li, Z. (2014). Locality of connective constants, I. Transitive graphs. Available at arXiv:1412.0150.
  • [11] Grimmett, G. R. and Li, Z. (2015). Locality of connective constants, II. Cayley graphs. Available at arXiv:1501.00476.
  • [12] Grimmett, G. R. and Marstrand, J. M. (1990). The supercritical phase of percolation is well behaved. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 430 439–457.
  • [13] Lyons, R. and Peres, Y. (2017). Probability on Trees and Networks. Cambridge Univ. Press, Cambridge.
  • [14] Montanari, A., Mossel, E. and Sly, A. (2012). The weak limit of Ising models on locally tree-like graphs. Probab. Theory Related Fields 152 31–51.
  • [15] Tassion, V. (2016). Crossing probabilities for Voronoi percolation. Ann. Probab. 44 3385–3398.