The Annals of Probability

Phase separation in random cluster models II: The droplet at equilibrium, and local deviation lower bounds

Alan Hammond

Full-text: Open access


We study the droplet that results from conditioning the planar subcritical Fortuin–Kasteleyn random cluster model on the presence of an open circuit Γ0 encircling the origin and enclosing an area of at least (or exactly) n2. We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Γ0), this being the maximum distance from a point in the circuit Γ0 to the boundary  conv(Γ0) of the circuit’s convex hull; and in a longitudinal sense by what we term maximum facet length, namely, the length of the longest line segment of which the polygon  conv(Γ0) is formed. We prove that there exists a constant c > 0 such that the conditional probability that the normalised quantity n−1/3(log n)−2/3 MLR(Γ0) exceeds c tends to 1 in the high n-limit; and that the same statement holds for n−2/3(log n)−1/3 MFL(Γ0). To obtain these bounds, we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a Markov chain that resamples sections of the circuit boundary. We analyse the chain at equilibrium to prove the local roughness lower bounds. Alongside complementary upper bounds provided in [14], the fluctuations MLR(Γ0) and MFL(Γ0) are determined up to a constant factor.

Article information

Ann. Probab., Volume 40, Number 3 (2012), 921-978.

First available in Project Euclid: 4 May 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Phase separation local roughness Wulff construction random cluster model


Hammond, Alan. Phase separation in random cluster models II: The droplet at equilibrium, and local deviation lower bounds. Ann. Probab. 40 (2012), no. 3, 921--978. doi:10.1214/11-AOP646.

Export citation


  • [1] Aizenman, M. and Barsky, D. J. (1987). Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 489–526.
  • [2] Aizenman, M., Barsky, D. J. and Fernández, R. (1987). The phase transition in a general class of Ising-type models is sharp. J. Stat. Phys. 47 343–374.
  • [3] Alexander, K. S. (1998). On weak mixing in lattice models. Probab. Theory Related Fields 110 441–471.
  • [4] Alexander, K. S. (2001). Cube-root boundary fluctuations for droplets in random cluster models. Comm. Math. Phys. 224 733–781.
  • [5] Alexander, K. S. (2004). Mixing properties and exponential decay for lattice systems in finite volumes. Ann. Probab. 32 441–487.
  • [6] Beffara, V. and Duminil-Copin, H. (2010). The self-dual point of the 2d random-cluster model is critical above q = 1. Available at arXiv:1006.5073.
  • [7] Campanino, M., Ioffe, D. and Velenik, Y. (2008). Fluctuation theory of connectivities for subcritical random cluster models. Ann. Probab. 36 1287–1321.
  • [8] Cerf, R. (2000). Large deviations for three dimensional supercritical percolation. Astérisque 267 vi+177.
  • [9] Cerf, R. and Pisztora, Á. (2000). On the Wulff crystal in the Ising model. Ann. Probab. 28 947–1017.
  • [10] Darling, D. A. (1983). On the supremum of a certain Gaussian process. Ann. Probab. 11 803–806.
  • [11] Dobrushin, R., Kotecký, R. and Shlosman, S. (1992). Wulff Construction: A Global Shape from Local Interaction. Translations of Mathematical Monographs 104. Amer. Math. Soc., Providence, RI.
  • [12] Grimmett, G. (2006). The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 333. Springer, Berlin.
  • [13] Häggström, O. (2007). Problem solving is often a matter of cooking up an appropriate Markov chain. Scand. J. Stat. 34 768–780.
  • [14] Hammond, A. (2010). Phase separation in random cluster models I: Uniform upper bounds on local deviation. Available at arXiv:1001.1527.
  • [15] Hammond, A. (2011). Phase separation in random cluster models III: Circuit regularity. J. Stat. Phys. 142 229–276.
  • [16] Ioffe, D. and Schonmann, R. H. (1998). Dobrushin–Kotecký–Shlosman theorem up to the critical temperature. Comm. Math. Phys. 199 117–167.
  • [17] Kardar, M., Parisi, G. and Zhang, Y. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889–892.
  • [18] Laanait, L., Messager, A., Miracle-Solé, S., Ruiz, J. and Shlosman, S. (1991). Interfaces in the Potts model. I. Pirogov–Sinai theory of the Fortuin–Kasteleyn representation. Comm. Math. Phys. 140 81–91.
  • [19] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [20] Louchard, G. (1984). Kac’s formula, Levy’s local time and Brownian excursion. J. Appl. Probab. 21 479–499.
  • [21] Majumdar, S. N. and Comtet, A. (2005). Airy distribution function: From the area under a Brownian excursion to the maximal height of fluctuating interfaces. J. Stat. Phys. 119 777–826.
  • [22] Uzun, H. B. and Alexander, K. S. (2003). Lower bounds for boundary roughness for droplets in Bernoulli percolation. Probab. Theory Related Fields 127 62–88.
  • [23] van den Berg, J., Häggström, O. and Kahn, J. (2006). Some conditional correlation inequalities for percolation and related processes. Random Structures Algorithms 29 417–435.
  • [24] Wulff, G. (1901). Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Kristallflächen. Zeitschrift für Krystallographie und Mineralogie 34 449–530.