Probability Theory and Related Fields

, Volume 90, Issue 1, pp 111–148

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

  • David J. Barsky
  • Geoffrey R. Grimmett
  • Charles M. Newman

DOI: 10.1007/BF01321136

Cite this article as:
Barsky, D.J., Grimmett, G.R. & Newman, C.M. Probab. Th. Rel. Fields (1991) 90: 111. doi:10.1007/BF01321136


Renormalization arguments are developed and applied to independent nearest-neighbor percolation on various subsets ℕ of ℤd,d≧2, yielding:
  • Equality of the critical densities,pc(ℕ), for ℕ 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,θ(p), atp=pc(ℕ).

Corollaries of these results include uniqueness of the infinite cluster for such ℕ'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.

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • David J. Barsky
    • 1
  • Geoffrey R. Grimmett
    • 2
  • Charles M. Newman
    • 3
  1. 1.Department of MathematicsUniversity of CaliforniaDavisUSA
  2. 2.School of MathematicsUniversity of BristolBristolEngland, UK
  3. 3.Department of MathematicsUniversity of ArizonaTucsonUSA