Summary
For two-dimensional Bernoulli percolation at densityp above the critical point, there exists a natural normg determined by the rate of decay of the connectivity function in every direction. IfW is the region of unit area with boundary of minimum possibleg-length, then it is known [4] that asN→∞, with probability approaching 1, conditionally onN≦|C(0)|<∞, the clusterC(0) of the origin approximatesW in shape to within a factor of 1±η(N) for some η(N)→0. Here a bound is established for the size η(N) of the fluctuations. Other types of conditioning which result in the formation of a shape approximatingW are also considered.
This is related to the quadratic stability of the variational minimum achieved by the Wulff curve ∂W: for somek>0, if γ is a curve enclosing a region of unit area such that the Hausdorff distanced H (γ+v,dW)≧δ for every translate γ+v, then theg-lengthg(γ) ≧g(∂W) +k δ2, at least for δ small.
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Research supported by NSF grant number DMS-8702906
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Alexander, K.S. Stability of the Wulff minimum and fluctuations in shape for large finite clusters in two-dimensional percolation. Probab. Th. Rel. Fields 91, 507–532 (1992). https://doi.org/10.1007/BF01192068
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DOI: https://doi.org/10.1007/BF01192068