  For a family of bond percolation models on 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.
