L. Chayes

The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation

AbstractWe consider two-dimensional Bernoulli percolation at densityp>pc and establish the following results:1.The probability,PN(p), that the origin is in afinite cluster of sizeN obeys

The Analysis of the Widom-Rowlinson Model by Stochastic Geometric Methods

\mathop {\lim }\limits_{N \to \infty } \frac{1}{{\sqrt N }}\log P_N (p) = - \frac{{\omega (p)\sigma (p)}}{{\sqrt {P_\infty (p)} }},

The inverse problem in classical statistical mechanics

whereP∞(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 asp↓pc. Roughly speaking, ω(p)σ(p) is the minimum surface energy of a “percolation droplet” of unit area.2.For all supercritical densitiesp>pc, the system obeys a microscopic Wulff construction: Namely, if the origin is conditioned to be in a finite cluster of sizeN, then with probability tending rapidly to 1 withN, the shape of this cluster-measured on the scale

The low-temperature behavior of disordered magnets

\sqrt N

Ornstein-Zernike Behavior for Self-Avoiding Walks at All Noncritical Temperatures

-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 withN, the excess sites in finite clusters will form a single large droplet, which-again on the scale