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Percolation in half-spaces: equality of critical densities and continuity of the percolation probability
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  • Published: March 1991

Percolation in half-spaces: equality of critical densities and continuity of the percolation probability

  • David J. Barsky1,
  • Geoffrey R. Grimmett2 &
  • Charles M. Newman3Ā 

Probability Theory and Related Fields volumeĀ 90,Ā pages 111–148 (1991)Cite this article

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Summary

Renormalization arguments are developed and applied to independent nearest-neighbor percolation on various subsets ā„• of ℤd,d≧2, yielding:

  • Equality of the critical densities,p c (ā„•), for ā„• a half-space, quarter-space, etc., and (ford>2) equality with the limit of slab critical densities.

  • Continuity of the phase transition for the half-space, quarter-space, etc.; i.e., vanishing of the percolation probability,Īø ā„•(p), atp=p c (ā„•).

Corollaries of these results include uniqueness of the infinite cluster for such ā„•'s and sufficiency of the following for proving continuity of the full-space phase transition: showing that percolation in the full-space at densityp implies percolation in the half-space at thesame density.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, 95616, Davis, CA, USA

    David J. Barsky

  2. School of Mathematics, University of Bristol, BS8 1TW, Bristol, England, UK

    Geoffrey R. Grimmett

  3. Department of Mathematics, University of Arizona, 85721, Tucson, AZ, USA

    Charles M. Newman

Authors
  1. David J. Barsky
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  2. Geoffrey R. Grimmett
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  3. Charles M. Newman
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Additional information

Research supported in part by an NSF Postdoctoral Fellowship (D.J.B.), the University of Arizona Center for the Study of Complex Systems (G.R.G.), NSF Grant DMS-8514834 and DMS-8902516 (C.M.N.), and AFOSR Contract No. F49620-86-C0130 to the Arizona Center for Mathematical Sciences under the U.R.I. Program

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Barsky, D.J., Grimmett, G.R. & Newman, C.M. Percolation in half-spaces: equality of critical densities and continuity of the percolation probability. Probab. Th. Rel. Fields 90, 111–148 (1991). https://doi.org/10.1007/BF01321136

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  • Received: 16 October 1989

  • Revised: 10 April 1991

  • Issue Date: March 1991

  • DOI: https://doi.org/10.1007/BF01321136

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Keywords

  • Phase Transition
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Critical Density
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