00-19 Alexander, K. S.

Cube-Root Boundary Fluctuations for Droplets in Random Cluster Models (199K, AMS-LATeX 1.2) Jan 13, 00

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Abstract. For a family of bond percolation models on $\mathbb{Z}^{2}$ that includes the Fortuin-Kasteleyn 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, Koteck\'y 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 $l = \sqrt{A}$ 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.

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