Summary
We consider thed-dimensional Bernoulli bond percolation model and prove the following results for allp<p c : (1) The leading power-law correction to exponential decay of the connectivity function between the origin and the point (L, 0, ..., 0) isL −(d−1)/2. (2) The correlation length, ξ(p) is real analytic. (3) Conditioned on the existence of a path between the origin and the point (L, 0, ..., 0), the hitting distribution of the cluster in the intermediate planes,x 1 =qL,0<q<1, obeys a multidimensional local limit theorem. Furthermore, for the two-dimensional percolation system, we prove the absence of a roughening transition: For allp>p c , the finite-volume conditional measures, defined by requiring the existence of a dual path between opposing faces of the boundary, converge—in the infinite-volume limit—to the standard Bernoulli measure.
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Work supported in part by G.N.A.F.A. (C.N.R.)
Work supported in part by NSF Grant No. DMS-88-06552
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Campanino, M., Chayes, J.T. & Chayes, L. Gaussian fluctuations of connectivities in the subcritical regime of percolation. Probab. Th. Rel. Fields 88, 269–341 (1991). https://doi.org/10.1007/BF01418864
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DOI: https://doi.org/10.1007/BF01418864