### Abstract

Renormalization arguments are developed and applied to independent nearest-neighbor percolation on various subsets ℕ of ℤ^{d}, d≧2, yielding:Equality of the critical densities, p_{c}(ℕ), for ℕ a half-space, quarter-space, etc., and (for d>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_{ℕ}(p), at p=p_{c}(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 density p implies percolation in the half-space at the same density.

Original language | English (US) |
---|---|

Pages (from-to) | 111-148 |

Number of pages | 38 |

Journal | Probability Theory and Related Fields |

Volume | 90 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1991 |

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### ASJC Scopus subject areas

- Statistics and Probability
- Analysis
- Mathematics(all)

### Cite this

*Probability Theory and Related Fields*,

*90*(1), 111-148. https://doi.org/10.1007/BF01321136

**Percolation in half-spaces : equality of critical densities and continuity of the percolation probability.** / Barsky, David J.; Grimmett, Geoffrey R.; Newman, Charles M.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 90, no. 1, pp. 111-148. https://doi.org/10.1007/BF01321136

}

TY - JOUR

T1 - Percolation in half-spaces

T2 - equality of critical densities and continuity of the percolation probability

AU - Barsky, David J.

AU - Grimmett, Geoffrey R.

AU - Newman, Charles M.

PY - 1991/3

Y1 - 1991/3

N2 - 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 (for d>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, θℕ(p), at p=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 density p implies percolation in the half-space at the same density.

AB - 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 (for d>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, θℕ(p), at p=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 density p implies percolation in the half-space at the same density.

UR - http://www.scopus.com/inward/record.url?scp=0000280765&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000280765&partnerID=8YFLogxK

U2 - 10.1007/BF01321136

DO - 10.1007/BF01321136

M3 - Article

AN - SCOPUS:0000280765

VL - 90

SP - 111

EP - 148

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 1

ER -