 0019 Alexander, K. S.
 CubeRoot Boundary Fluctuations for Droplets in Random Cluster Models
(199K, AMSLATeX 1.2)
Jan 13, 00

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Abstract. For a family of bond percolation models on $\mathbb{Z}^{2}$
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, 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 longrange. 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 longrange part of the deviation from
a Wulff shape, and we establish the absense of ``bottlenecks,'' which
are a form of selfapproach 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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