Summary
We extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent percolation to cover random graphs on ℤd or ℤd × ℕ with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on ℤd or on a slab ℤd × {0,...K} and prove both that percolation occurs and that the infinite component is unique forV=ℤ2×{0,1} or larger.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aizenman, M., Newman, C.M., Kesten, H.: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Commun. Math. Phys.111, 505–531 (1987)
Barsky, D.J., Grimmett, G.R., Newman, C.M.: Percolation in half spaces: equality of critical densities and continuity of the percolation probability. Probab. Theory Relat. Fields90, 111–148 (1991)
Bollobás, B.: Random graphs. London: Academic Press 1985
Burton, R.M., Keane, M.: Density and uniqueness in percolation. Commun. Math. Phys.121, 501–505 (1989)
Campanino, M., Russo, L.: An upper bound for the critical probability for the three-dimensional cubic lattice. Ann. Probab.13, 478–491 (1985)
Coniglio, A., Nappi, C.R., Peruggi, F., Russo, L.: Percolation and phase transition in the Ising model. Commun. Math. Phys.51, 315–323 (1976)
Durrett, R., Kesten, H.: The critical parameter for connectedness of some random graphs. In: Baker, A., Bollobas, B., Hajnal, A. (eds.) A tribute to Paul Erdös, pp. 161–176. Cambridge: Cambridge University Press 1990
Edwards, R.G., Sokal, A.D.: Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D38, 2009–2012 (1988)
Erdös, P., Rényi, A.: On random graphs I. Publ. Math.6, 290–297 (1959)
Fisher, M.E.: Critical probabilities for cluster size and percolation problems. J. Math. Phys.2, 620–627 (1961)
Fortuin, C.M.: On the random-cluster model III. The simple random-cluster model. Physica59, 545–570 (1972)
Fortuin, C.M., Kasteleyn, P.W.: On the random-cluster model. I. Introduction and relation to other models. Physica57, 536–564 (1972)
Gandolfi, A.: Uniqueness of the infinite cluster for stationary Gibbs states. Ann. Probab.17, 1403–1415 (1989)
Gandolfi, A., Grimmett, G., Russo, L.: On the uniqueness of the infinite cluster in the percolation model. Commun. Math. Phys.114, 549–552 (1988a)
Gandolfi, A., Keane, M.S., Russo, L.: On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation. Ann. Probab.16 (3), 1147–1157 (1988b)
Grimmett, G.R., Keane, M.S., Marstand, J.M.: On the connectedness of a random graph. Math. Proc. Camb. Philos. Soc.96, 151–166 (1984)
Grimmett, G.R., Newman, C.M.: Percolation in ∞+1 dimensions. In: Grimmett, G., Welsh, D. (eds.). Hammersley Festschrift. Oxford University Press 1988
Harris, T.E.: A lower bound for the critical probability in a certain percolation process. Proc. Camb. Philos. Soc.56, 13–20 (1960)
Kalikow, S., Weiss, B.: When are random graphs connected. (1988) preprint.
Kasai, Y., Okiji, A.: Percolation problem describing±J Ising spin glass system. Prog. Theor. Phys.79, 1080–1094 (1988)
Kasteleyn, P.W., Fortuin, C.M.: Phase transitions in lattice systems with random local properties. J. Phys. Soc. Japan26, 11–14 (1969)
Kesten, H.: Percolation theory for mathematicians. Boston Basel Stuttgart: Birkhäuser 1982
Kesten, H.: Correlation length and critical probabilities of slabs for percolation. (Preprint, 1988)
Kesten, H.: Connectivity of certain graphs on halfspaces, quarter spaces. Proceedings Probability Conference Singapore 1989 (to appear)
Meester, R.W.J.: An algorithm for calculating critical probabilities and percolation functions in percolation models defined by rotations. Ergodic Theory Dyn. Syst.9, 495–509 (1989)
Newman, C.M.: Ising models and dependent percolation. In: Block, H.W., Sampson, A.R., Savits, T.H. (eds.) Topics in statistical dependence (IMS Lect. Notes—Monograph Series, vol. 16, pp. 395–401
Newman, C.M., Schulman, L.S.: Infinite clusters in percolation models. J. Stat. Phys.26, (3) 613–628 (1981)
Russo, L.: An approximate zero-one law. Z. Wahrscheinlichkeitstheor. Verw. Geb.61, 129–139 (1982)
Shepp, L.A.: Connectedness of certain random graphs. Jsr. J. Math.67, 23–33 (1989)
Swendsen, R.H., Wang, J.S.: Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett.58, 86–88 (1987)
Author information
Authors and Affiliations
Additional information
A.G. was partially supported from AFOSR through grant no. 90-0090
Rights and permissions
About this article
Cite this article
Gandolfi, A., Keane, M.S. & Newman, C.M. Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probab. Th. Rel. Fields 92, 511–527 (1992). https://doi.org/10.1007/BF01274266
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01274266