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

Abstract

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. © 1991 Springer-Verlag.