Phase Separation in Random Cluster Models I: Uniform Upper Bounds on Local Deviation

Abstract

This is the first in a series of three papers that addresses the behaviour of the droplet that results, in the percolating phase, from conditioning the planar Fortuin-Kasteleyn random cluster model on the presence of an open dual circuit Γ0 encircling the origin and enclosing an area of at least (or exactly) n2. (By the Fortuin-Kasteleyn representation, the model is a close relative of the droplet formed by conditioning the Potts model on an excess of spins of a given type.) 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 ∂ of the circuit’s convex hull; and in a longitudinal sense by what we term maximum facet length, MLF(Γ0), namely, the length of the longest line segment of which the polygon ∂ is formed. The principal conclusion of the series of papers is the following uniform control on local deviation: that there are constants 0 < c < C < ∞ such that the conditional probability that the normalized quantity n−1/3(log n )−2/3MLR lies in the interval [c, C] tends to 1 in the high n-limit; and that the same statement holds for n−2/3 (log n )−1/3 MLF. In this way, we confirm the anticipated n1/3 scaling of maximum local roughness, and provide a sharp logarithmic power-law correction. This local deviation behaviour occurs by means of locally Gaussian effects constrained globally by curvature, and we believe that it arises in many radially defined stochastic interface models, including growth models belonging to the Kardar-Parisi-Zhang universality class.