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Gaussian fluctuations of connectivities in the subcritical regime of percolation
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  • Published: September 1991

Gaussian fluctuations of connectivities in the subcritical regime of percolation

  • Massimo Campanino1,
  • J. T. Chayes2 &
  • L. Chayes2 

Probability Theory and Related Fields volume 88, pages 269–341 (1991)Cite this article

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  • 28 Citations

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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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Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, I-40126, Bologna, Italy

    Massimo Campanino

  2. Department of Mathematics, University of California, 90024, Los Angeles, CA, USA

    J. T. Chayes & L. Chayes

Authors
  1. Massimo Campanino
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  2. J. T. Chayes
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  3. L. Chayes
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Additional information

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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  • Received: 30 November 1988

  • Revised: 08 June 1990

  • Issue Date: September 1991

  • DOI: https://doi.org/10.1007/BF01418864

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Keywords

  • Mathematical Biology
  • Correlation Length
  • Local Limit
  • Percolation Model
  • Connectivity Function
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