Cube-Root Boundary Fluctuations¶for Droplets in Random Cluster Models

Abstract

Abstract. For a family of bond percolation models on Z 2 that includes the FortuinKasteleyn random cluster model, we consider properties of the "droplet" that results, in the percolating regime, from conditioning on the existence of an open dual circuit surrounding the origin and enclosing at least (or exactly) a given large area A. This droplet is a close surrogate for the one obtained by Dobrushin, Kotecký and Shlosman by conditioning the Ising model; it approximates an area-A Wulff shape. The local part of the deviation from the Wulff shape (the "local roughness") is the inward deviation of the droplet boundary from the boundary of its own convex hull; the remaining part of the deviation, that of the convex hull of the droplet from the Wulff shape, is inherently long-range. We show that the local roughness is described by at most the exponent 1/3 predicted by nonrigorous theory; this same prediction has been made for a wide class of interfaces in two dimensions. Specifically, the average of the local roughness over the droplet surface is shown to be O(l 1/3 (log l) 2/3 ) in probability, where l = √ A is the linear scale of the droplet. We also bound the maximum of the local roughness over the droplet surface and bound the long-range part of the deviation from a Wulff shape, and we establish the absense of "bottlenecks," which are a form of self-approach by the droplet boundary, down to scale log l. Finally, if we condition instead on the event that the total area of all large droplets inside a finite box exceeds A, we show that with probability near 1 for large A, only a single large droplet is present.

Lower bounds for boundary roughness for droplets in Bernoulli percolation

“…The next lemma gives a lower bound on the sum of the lengths of long sides when diam( 0 ) is not abnormally large. From [Al3], for some K 9 , K 10 , K 11 > 0, for T > 0,…”

“…We will use the coarse graining setup and results of [Al3]. For s > 0, and any contour with a τ -diameter of at least 2s, the coarse graining algorithm selects a subset {w 0 , w 1 , · · · , w m+1 } of the extreme points of Co(γ ), with w m+1 = w 0 , called the s-hull skeleton of γ and denoted HSkel s (γ ).…”

“…We denote the polygonal path w 0 → w 1 → · · · → w m+1 by HPath s (γ ). The specifics of the algorithm for choosing the s-hull skeleton are not important to us here; we refer the reader to [Al3]. What we need are the following properties, also from [Al3].…”

“…Moreover, they showed that the Hausdorff distance between the droplet boundary γ and the boundary of the Wulff shape W is bounded by a power of the linear scale of the droplet. This Hausdorff distance is related but not equivalent to local roughness; see [Al3]. The very-low-temperature restriction was removed by Ioffe and Schonmann [IS], who proved Dobrushin-Kotecky-Schlosman theorem up to the critical temperature.…”

Stochastic Interface Models

“…A plaquette is a closed loop P = {b (1) , b (2) , b (3) , b (4) Notice that the condition (P1) follows from (L) by taking the closed loop C = {b, −b}. We set…”