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.
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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 -