Abstract
We consider two-dimensional Bernoulli percolation at densityp>p c and establish the following results:
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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.
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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.
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Communicated by M. Aizenman
Work supported in part by NSF Grant No. DMS-87-02906
Work supported in part by NSF Grant No. DMS-88-06552
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Alexander, K., Chayes, J.T. & Chayes, L. The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation. Commun.Math. Phys. 131, 1–50 (1990). https://doi.org/10.1007/BF02097679
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DOI: https://doi.org/10.1007/BF02097679