STABILITY OF THE WULFF MINIMUM AND FLUCTUATIONS IN SHAPE FOR LARGE FINITE CLUSTERS IN 2-DIMENSIONAL PERCOLATION

For two-dimensional Bernoulli percolation at density p above the critical point, there exists a natural norm g determined by the rate of decay of the connectivity function in every direction. If W is the region of unit area with boundary of minimum possible g-length, then it is known [4] that as N --> infinity, with probability approaching 1, conditionally on N less-than-or-equal-to \C(0)\ < infinity, the cluster C(0) of the origin approximates W in shape to within a factor of 1 +/- eta(N) for some eta(N) --> 0. Here a bound is established for the size eta(N) of the fluctuations. Other types of conditioning which result in the formation of a shape approximating W are also considered. This is related to the quadratic stability of the variational minimum achieved by the Wulff curve partial-derivative W: for some k > 0, if gamma is a curve enclosing a region of unit area such that the Hausdorff distance d(H)(gamma + upsilon, dW) greater-than-or-equal-to delta for every translate gamma + upsilon, then the g-length g(gamma) greater-than-or-equal-to g (partial-derivative W) + k-delta(2), at least for delta-small.