, Volume 90, Issue 1, pp 111-148

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

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Summary

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 (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,θ (p), atp=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 densityp implies percolation in the half-space at thesame density.

    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