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The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation. (English) Zbl 0698.60098

Summary: We consider two-dimensional Bernoulli percolation at density \(p>p_ c\) and establish the following results:
1. The probability, \(P_ N(p)\), that the origin is in a finite cluster of size N obeys \[ \lim_{N\to \infty}N^{-1/2} \log P_ N(p)=-\omega (p)\sigma (p)/\sqrt{P_{\infty}(p)}, \] where \(P_{\infty}(p)\) is the infinite cluster density, \(\sigma(p)\) is the (zero-angle) surface tension, and \(\omega(p)\) is a quantity which remains positive and finite as \(p\downarrow p_ c\). Roughly speaking, \(\omega(p)\sigma(p)\) is the minimum surface energy of a “percolation droplet” of unit area.
2. For all supercritical densities \(p>p_ c\), the systen 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 \(\sqrt{N}\)- 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 with N, 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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