Summary
Renormalization arguments are developed and applied to independent nearest-neighbor percolation on various subsets ℕ of ℤd,d≧2, yielding:
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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.
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Continuity of the phase transition for the half-space, quarter-space, etc.; i.e., vanishing of the percolation probability,θ ℕ(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.
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References
Aizenman, M., Newman, C.M.: Discontinuity of the percolation density in one dimensional 1/|x−y|2 percolation models. Commun. Math. Phys.107, 611–647 (1986)
Aizenman, M., Chayes, J.T., Chayes, L., Fröhlich, J., Russo, L.: On a sharp transition from area law to perimeter law in a system of random surfaces. Commun. Math. Phys.92, 19–69 (1983)
Aizenman, M., Kesten, H., Newman, C.M.: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Commun. Math. Phys.111, 505–532 (1987)
Barsky, D.J.: Critical points and critical exponents in percolation and Ising-type models. Ph.D. dissertation. Rutgers University, 1987
Barsky, D.J., Aizenman, M.: Percolation critical exponents under the triangle condition. Ann. Probab. (in press 1991)
Barsky, D.J., Grimmett, G.R., Newman, C.M.: Dynamic renormalization and continuity of the percolation transition in orthants. In: Alexander, K., Watkins, J. (eds.) Spatial stochastic processes. Boston: Birkhäuser 1989
Berg van den, J., Keane, M.: On the continuity of the percolation probability function. In: Beals, R., Beck, A., Bellow, A., Hajain, A. (eds) Conference in modern analysis and probability. Contemp. Math.26, 61–65 (1984)
Bezuidenhout, C.E., Grimmett, G.R.: The critical contact process dies out. Ann. Probab.18, 1462–1482 (1990)
Burton, R., Keane, M.: Density and uniqueness in percolation. Commun. Math. Phys.121, 501–506 (1989)
Chayes, J.T., Chayes, L.: Critical points and intermediate phases on wedges of ℤd. J. Phys. A19, 3033–3048 (1986)
Durrett, R.: Oriented percolation in two dimensions. Ann. Probab.12, 999–1040 (1984)
Fisher, M.E.: Critical probabilities for cluster size and percolation problems. J. Math. Phys.2, 620–627 (1961)
Gandolfi, A.: Uniqueness of the infinite cluster for stationary finite range Gibbs states. Ann. Probab.17, 1403–1415 (1989)
Gandolfi, A., Grimmett, G.R., Russo, L.: On the uniqueness of the infinite open cluster in the percolation model. Commun. Math. Phys.114, 549–552 (1988)
Gandolfi, A., Keane, M., Newman, C.M.: Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Submitted to Probab. Th. Rel. Fields (1991)
Grimmett, G.R.: Percolation. Berlin Heidelberg New York: Springer 1989
Grimmett, G.R., Marstrand, J.M.: The supercritical phase of percolation is well behaved. Proc. R. Soc. Lond. Ser. A430, 439–457 (1990)
Hara, T., Slade, G.: The mean field critical behavior of percolation in high dimensions. In: Simon, B., Truman, A., Davies, I.M. (eds.) IXth International Congress on Mathematical Physics. Bristol: Adam Hilger 450–453 (1989)
Hara, T., Slade, G.: Mean field critical behavior for percolation in high dimensions. Commun. Math. Phys.128, 333–391 (1990)
Harris, T.E.: A lower bound for the critical probability in a certain percolation process. Proc. Camb. Phil. Soc.56, 13–20 (1960)
Kesten, H.: The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys.74, 41–59 (1980)
Kesten, H.: Scaling relations for 2D-percolation. Commun. Math. Phys.109, 109–156 (1987)
Kesten, H.: Correlation length and critical probabilities of slabs for percolation. Preprint, Cornell University (1989)
Newman, C.M.: Some critical exponent inequalities for percolation. J. Stat. Phys.45, 359–368 (1986)
Newman, C.M., Schulman, L.S.: One-dimensional 1/|j−i|s percolation models: the existence of a transition fors≦2. Commun. Math. Phys.104, 547–571 (1986)
Russo, L.: A note on percolation. Z. Wahrscheinlichkeitstheor. Verw. Geb.43, 39–48 (1978)
Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 229–237 (1981)
Seymour, P.D., Welsh, D.J.A.: Percolation probabilities on the square lattice. In: Bollobas, B. (ed.) Advances in graph theory. Ann. Discrete Math.3, 227–245 (1978)
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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, 111–148 (1991). https://doi.org/10.1007/BF01321136
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DOI: https://doi.org/10.1007/BF01321136