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