Summary
Renormalization arguments are developed and applied to independent nearest-neighbor percolation on various subsets ā of ā¤d,dā§2, yielding:
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Equality of the critical densities,p c (ā), for ā a half-space, quarter-space, etc., and (ford>2) equality with the limit of slab critical densities.
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Continuity of the phase transition for the half-space, quarter-space, etc.; i.e., vanishing of the percolation probability,Īø ā(p), atp=p c (ā).
Corollaries of these results include uniqueness of the infinite cluster for such ā'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.
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Research supported in part by an NSF Postdoctoral Fellowship (D.J.B.), the University of Arizona Center for the Study of Complex Systems (G.R.G.), NSF Grant DMS-8514834 and DMS-8902516 (C.M.N.), and AFOSR Contract No. F49620-86-C0130 to the Arizona Center for Mathematical Sciences under the U.R.I. Program
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Barsky, D.J., Grimmett, G.R. & Newman, C.M. Percolation in half-spaces: equality of critical densities and continuity of the percolation probability. Probab. Th. Rel. Fields 90, 111ā148 (1991). https://doi.org/10.1007/BF01321136
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DOI: https://doi.org/10.1007/BF01321136
Keywords
- Phase Transition
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Critical Density