ISSN:
1432-0916
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract We consider two-dimensional Bernoulli percolation at densityp〉p c and establish the following results: 1. The probability,P N (p), that the origin is in afinite cluster of sizeN obeys $$\mathop {\lim }\limits_{N \to \infty } \frac{1}{{\sqrt N }}\log P_N (p) = - \frac{{\omega (p)\sigma (p)}}{{\sqrt {P_\infty (p)} }},$$ 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↓p c . Roughly speaking, ω(p)σ(p) is the minimum surface energy of a “percolation droplet” of unit area. 2. For all supercritical densitiesp〉p c , 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 $$\sqrt N$$ -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 $$\sqrt N$$ -will assume the classical Wulff shape.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02097679
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