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

TL;DR

This paper investigates the geometric fluctuations of droplets in the subcritical Fortuin-Kasteleyn random cluster model, establishing lower bounds on boundary roughness and facet length that match known upper bounds, revealing precise fluctuation scales.

Contribution

It introduces a Markov chain resampling approach to analyze boundary fluctuations of droplets conditioned on high area, providing sharp lower bounds matching previous upper bounds.

Findings

Boundary roughness scales as n^{1/3} with logarithmic corrections.

Facet length scales as n^{2/3} with logarithmic corrections.

Fluctuation scales are determined up to a constant factor.

Abstract

We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn planar random cluster model on the presence of an open circuit Gamma_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(Gamma_0), this being the maximum distance from a point in the circuit Gamma_0 to the boundary of the circuit's convex hull; and in a longitudinal sense by what we term maximum facet length, MFL(Gamma_0), namely, the length of the longest line segment of which the boundary of the convex hull is formed. We prove that that there exists a constant c > 0 such that the conditional probability that the normalised quantity n^{-1/3}\big(\log n \big)^{-2/3} MLR(Gamma_0) exceeds c tends to 1 in the high n-limit; and that the same statement holds for n^{-2/3}\big(\log n \big)^{-1/3} MFL(Gamma_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 arXiv:1001.1527, the fluctuations MLR(Gamma_0) and MFL(Gamma_0) are determined up to a constant factor.