Lower bounds for boundary roughness for droplets in Bernoulli percolation

We consider boundary roughness for the ``droplet'' created when supercritical two-dimensional Bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least l2, for large l. The maximum local roughness is the maximum inward deviation of the droplet boundary from the boundary of its own convex hull; we show that for large l this maximum is at least of order l1/3(logl)−2/3. This complements the upper bound of order l1/3(logl)−2/3 proved in [Al3] for the average local roughness. The exponent 1/3 on l here is in keeping with predictions from the physics literature for interfaces in two dimensions.