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 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.