Lower bounds for boundary roughness for droplets in Bernoulli percolation

We consider boundary roughness for the "dropletʹʹ created when supercritical two-dimensional Bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least l^2, 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 l^(1/3)(logl^(–2/3). This complements the upper bound of order l^(1/3)(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.