We study the droplet that results from conditioning the planar subcritical Fortuin–Kasteleyn random cluster model on the presence of an open circuit Γ₀ encircling the origin and enclosing an area of at least (or exactly) n². We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Γ₀), this being the maximum distance from a point in the circuit Γ₀ to the boundary ∂ conv(Γ₀) of the circuit’s convex hull; and in a longitudinal sense by what we term maximum facet length, namely, the length of the longest line segment of which the polygon ∂ conv(Γ₀) is formed. We prove that there exists a constant c > 0 such that the conditional probability that the normalised quantity n^−1/3(log n)^−2/3 MLR(Γ₀) exceeds c tends to 1 in the high n-limit; and that the same statement holds for n^−2/3(log n)^−1/3 MFL(Γ₀). 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 [14], the fluctuations MLR(Γ₀) and MFL(Γ₀) are determined up to a constant factor.

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