Electronic Journal of Probability

Intrinsic isoperimetry of the giant component of supercritical bond percolation in dimension two

Julian Gold

Full-text: Open access


We study the isoperimetric subgraphs of the giant component $\mathbf{{C}} _n$ of supercritical bond percolation on the square lattice. These are subgraphs of $\mathbf{{C}} _n$ with minimal edge boundary to volume ratio. In contrast to the work of [8], the edge boundary is taken only within $\mathbf{{C}} _n$ instead of the full infinite cluster. The isoperimetric subgraphs are shown to converge almost surely, after rescaling, to the collection of optimizers of a continuum isoperimetric problem emerging naturally from the model. We also show that the Cheeger constant of $\mathbf{{C}} _n$ scales to a deterministic constant, which is itself an isoperimetric ratio, settling a conjecture of Benjamini in dimension two.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 53, 41 pp.

Received: 14 May 2017
Accepted: 15 May 2018
First available in Project Euclid: 1 June 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35] 52B60: Isoperimetric problems for polytopes

percolation Cheeger constant isoperimetry

Creative Commons Attribution 4.0 International License.


Gold, Julian. Intrinsic isoperimetry of the giant component of supercritical bond percolation in dimension two. Electron. J. Probab. 23 (2018), paper no. 53, 41 pp. doi:10.1214/18-EJP178. https://projecteuclid.org/euclid.ejp/1527818431

Export citation


  • [1] K. Alexander, J.T. Chayes, and L. Chayes, The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional bernoulli percolation, Comm. Math. Phys 131 (1990), no. 1, 1–50.
  • [2] N. Alon, Eigenvalues and expanders, Combinatorica 6 (1986), no. 2, 83–96.
  • [3] N. Alon and V. D. Milman, $\lambda _1,$ Isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B 38 (1985), no. 1, 73–88.
  • [4] L. Ambrosio, V. Caselles, S. Masnou, and J.-M. Morel, Connected components of sets of finite perimeter and applications to image processing, J. Eur. Math. Soc. (JEMS) 3 (2001), no. 1, 39–92.
  • [5] A. Auffinger, M. Damron, and J. Hanson, 50 years of first-passage percolation, vol. 68, American Mathematical Soc., 2017.
  • [6] I. Benjamini and E. Mossel, On the mixing time of a simple random walk on the super critical percolation cluster, Probab. Theory Related Fields 125 (2003), no. 3, 408–420.
  • [7] N. Berger, M. Biskup, C. E. Hoffman, and G. Kozma, Anomalous heat-kernel decay for random walk among bounded random conductances, Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), no. 2, 374–392.
  • [8] M. Biskup, O. Louidor, E. B. Procaccia, and R. Rosenthal, Isoperimetry in two-dimensional percolation, Comm. Pure Appl. Math. 68 (2015), no. 9, 1483–1531.
  • [9] T. Bodineau, The Wulff construction in three and more dimensions, Comm. Math. Phys. 207 (1999), no. 1, 197–229.
  • [10] T. Bodineau, On the van der Waals theory of surface tension, Markov Process. Related Fields 8 (2002), no. 2, 319–338.
  • [11] T. Bodineau, D. Ioffe, and Y. Velenik, Rigorous probabilistic analysis of equilibrium crystal shapes, J. Math. Phys. 41 (2000), no. 3, 1033–1098.
  • [12] T. Bodineau, D. Ioffe, and Y. Velenik, Winterbottom construction for finite range ferromagnetic models: an $\mathbb L_1$-approach, J. Statist. Phys. 105 (2001), no. 1–2, 93–131.
  • [13] R. Cerf, Large deviations for three dimensional supercritical percolation, Astérisque (2000), no. 267, vi–177.
  • [14] R. Cerf, The Wulff crystal in Ising and percolation models, Lecture Notes in Mathematics, vol. 1878, Springer-Verlag, Berlin, 2006.
  • [15] R. Cerf and Á. Pisztora, On the Wulff crystal in the Ising model, Ann. Probab. 28 (2000), no. 3, 947–1017.
  • [16] R. Cerf and Á. Pisztora, Phase coexistence in Ising, Potts and percolation models, Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 6, 643–724.
  • [17] R. Cerf and M. Théret, Maximal stream and minimal cutset for first passage percolation through a domain of $\mathbb{R} ^d$, Ann. Probab. 42 (2012), no. 3, 1054–1120.
  • [18] J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Proceedings of the Princeton conference in honor of Professor S. Bochner, Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199.
  • [19] F. R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, vol. 92, Conference Board of the Mathematical Sciences, Washington, D.C., 1997.
  • [20] R. L. Dobrushin, R. Kotecký, and S.B. Shlosman, Wulff construction: a global shape from local interaction, vol. 104, American Mathematical Society Providence, Rhode Island, 1992.
  • [21] R. Durrett and R. H. Schonmann, Large deviations for the contact process and two-dimensional percolation, Probab. Theory Related Fields 77 (1988), no. 4, 583–603.
  • [22] O. Garet, R. Marchand, E. B. Procaccia, and M. Théret, Continuity of the time and isoperimetric constants in supercritical percolation, Electron. J. Probab. 22 (2017), 78–113.
  • [23] J. W. Gibbs, On the equilibrium of heterogeneous substances, American Journal of Science (1878), no. 96, 441–458.
  • [24] J. Gold, Isoperimetry in supercritical bond percolation in dimensions three and higher, to appear Ann. Inst. H. Poincaré Probab. Statist.,arxiv:1602.05598 (2017).
  • [25] G. Grimmett, Percolation, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, Springer-Verlag, Berlin, 1999.
  • [26] H. Kesten, The critical probability of bond percolation on the square lattice equals $1\over 2$, Comm. Math. Phys. 74 (1980), no. 1, 41–59.
  • [27] H. Kesten and Y. Zhang, The probability of a large finite cluster in supercritical Bernoulli percolation, Ann. Probab. 18 (1990), no. 2, 537–555.
  • [28] R. Kotecký and C. E. Pfister, Equilibrium shapes of crystals attached to walls, J. Stat. Phys. 76 (1994), no. 1–2, 419–445.
  • [29] F. Maggi, Sets of finite perimeter and geometric variational problems, Cambridge Studies in Advanced Mathematics, vol. 135, Cambridge University Press, Cambridge, 2012.
  • [30] P. Mathieu and E. Remy, Isoperimetry and heat kernel decay on percolation clusters, Ann. Probab. 32 (2004), no. 1A, 100–128.
  • [31] G. Pete, A note on percolation on $\mathbb Z^d$: isoperimetric profile via exponential cluster repulsion, Electron. Commun. Probab. 13 (2008), 377–392.
  • [32] C. E. Pfister and Y. Velenik, Mathematical theory of the wetting phenomenon in the 2d Ising model, Helv. Phys. Acta 69 (1996), 949–973.
  • [33] E. B. Procaccia and R. Rosenthal, Concentration estimates for the isoperimetric constant of the supercritical percolation cluster, Electron. Commun. Probab. 17 (2012), no. 30, 1–11.
  • [34] C. Rau, Sur le nombre de points visités par une marche aléatoire sur un amas infini de percolation, Bull. Soc. Math. France 135 (2007), no. 1, 135–169.
  • [35] S. B. Shlosman, The droplet in the tube: a case of phase transition in the canonical ensemble, Comm. Math. Phys. 125 (1989), no. 1, 81–90.
  • [36] R. M. Tanner, Explicit concentrators from generalized $N$-gons, SIAM J. Algebraic Discrete Methods 5 (1984), no. 3, 287–293.
  • [37] J. E. Taylor, Existence and structure of solutions to a class of nonelliptic variational problems, Sympos. Math. 14 (1974), no. 4, 499–508.
  • [38] J. E. Taylor, Unique structure of solutions to a class of nonelliptic variational problems, Proc. Sympos. Pure Math. 27 (1975), 419–427.
  • [39] J. E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc. 84 (1978), no. 4, 568–588.
  • [40] W. L. Winterbottom, Equilibrium shape of a small particle in contact with a foreign substrate, Acta Metall. 15 (1967), no. 2, 303–310.
  • [41] G. Wulff, Zur frage der geschwindigkeit des wachstums und der auflösung der kristallflachen, Z. Kryst. Miner 34 (1901), 449–530.