## The Annals of Probability

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

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

#### Abstract

We study the droplet that results from conditioning the planar subcritical Fortuin–Kasteleyn random cluster model on the presence of an open circuit Γ_{0} encircling the origin and enclosing an area of at least (or exactly) *n*^{2}. We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Γ_{0}), this being the maximum distance from a point in the circuit Γ_{0} to the boundary *∂* conv(Γ_{0}) 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(Γ_{0}) 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(Γ_{0}) 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(Γ_{0}). 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(Γ_{0}) and MFL(Γ_{0}) are determined up to a constant factor.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 4 May 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1336136055

**Digital Object Identifier**

doi:10.1214/11-AOP646

**Mathematical Reviews number (MathSciNet)**

MR2962083

**Zentralblatt MATH identifier**

1271.60021

**Subjects**

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

**Keywords**

Phase separation local roughness Wulff construction random cluster model

#### Citation

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. https://projecteuclid.org/euclid.aop/1336136055