THE WULFF CONSTRUCTION AND ASYMPTOTICS OF THE FINITE CLUSTER DISTRIBUTION FOR 2-DIMENSIONAL BERNOULLI PERCOLATION

We consider two-dimensional Bernoulli percolation at density p>pc and establish the following results: 1. The probability, PN(p), that the origin is in a finite cluster of size N obeys {Mathematical expression} where P∞(p) is the infinite cluster density, σ(p) is the (zero-angle) surface tension, and ω(p) is a quantity which remains positive and finite as p↓pc. Roughly speaking, ω(p)σ(p) is the minimum surface energy of a "percolation droplet" of unit area. 2. For all supercritical densities p>pc, the system obeys a microscopic Wulff construction: Namely, if the origin is conditioned to be in a finite cluster of size N, then with probability tending rapidly to 1 with N, the shape of this cluster-measured on the scale {Mathematical expression}-will be that predicted by the classical Wulff construction. Alternatively, if a system of finite volume, N, is restricted to a "microcanonical ensemble" in which the infinite cluster density is below its usual value, then with probability tending rapidly to 1 with N, the excess sites in finite clusters will form a single large droplet, which-again on the scale {Mathematical expression}-will assume the classical Wulff shape. © 1990 Springer-Verlag.