  We study the droplet that results from conditioning the subcritical
Fortuin-Kasteleyn planar random cluster model on the presence of an open
circuit Gamma_0 encircling the origin and enclosing an area of at least (or
exactly) n^2. We consider local deviation of the droplet boundary, measured in
a radial sense by the maximum local roughness, MLR(Gamma_0), this being the
maximum distance from a point in the circuit Gamma_0 to the boundary of the
circuit's convex hull; and in a longitudinal sense by what we term maximum
facet length, MFL(Gamma_0), namely, the length of the longest line segment of
which the boundary of the convex hull is formed. We prove that that there
exists a constant c > 0 such that the conditional probability that the
normalised quantity n^{-1/3}\big(\log n \big)^{-2/3} MLR(Gamma_0) exceeds c
tends to 1 in the high n-limit; and that the same statement holds for
n^{-2/3}\big(\log n \big)^{-1/3} MFL(Gamma_0). To obtain these bounds, we
exhibit the random cluster measure conditional on the presence of an open
circuit trapping high area as the invariant measure of a Markov chain that
resamples sections of the circuit boundary. We analyse the chain at equilibrium
to prove the local roughness lower bounds. Alongside complementary upper bounds
provided in arXiv:1001.1527, the fluctuations MLR(Gamma_0) and MFL(Gamma_0) are
determined up to a constant factor.
