Abstract
For independent translation-invariant irreducible percolation models, it is proved that the infinite cluster, when it exists, must be unique. The proof is based on the convexity (or almost convexity) and differentiability of the mean number of clusters per site, which is the percolation analogue of the free energy. The analysis applies to both site and bond models in arbitrary dimension, including long range bond percolation. In particular, uniqueness is valid at the critical point of one-dimensional 1/∣x−y∣2 models in spite of the discontinuity of the percolation density there. Corollaries of uniqueness and its proof are continuity of the connectivity functions and (except possibly at the critical point) of the percolation density. Related to differentiability of the free energy are inequalities which bound the “specific heat” critical exponent α in terms of the mean cluster size exponent γ and the critical cluster size distribution exponent δ; e.g., 1+α≦γ (δ/2−1)/(δ−1).
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Aizenman, M., Barsky, D. J.: Sharpness of the phase transition in percolation models, Commun. Math. Phys.108, 489–526 (1987).
Aizenman, M., Barsky, D. J.: in preparation. See also Barsky, D. J. Rutgers University Ph.D. thesis (1987).
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., Newman, C. M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys.36, 107–143 (1984)
Aizenman, M., Newman, C. M.: Discontinuity of the percolation density in one-dimensional 1/∣x−y∣2 percolation models. Commun. Math. Phys.107, 611–648 (1986)
van den Berg, J., Keane, M.: On the continuity of the percolation probability function, Contemp. Math.26, 61–65 (1984)
Bricmont, J., Lebowitz, J. L.: On the continuity of the magnetization and energy in Ising ferromagnets. J. Stat. Phys.42, 861–869 (1986)
Choquet, G.: Topology. New York: Academic Press 1966
Coniglio, A.: Shapes, surfaces, and interfaces in percolation clusters. In: Physics of finely divided matter. Daoud M. (ed.). Proc. Les Houches Conf. of March, 1985 (to appear)
Chayes, J. T., Chayes, L., Newman, C. M., The stochastic geometry of invasion percolation. Commun. Math. Phys.101, 383–407 (1985)
Durrett, R., Nguyen, B.: Thermodynamic inequalities for percolation. Commun. Math. Phys.99, 253–269 (1985)
Fisher, M. E.: Critical probabilities for cluster size and percolation problems. J. Math. Phys.2, 620–627 (1961)
Fortuin, C., Kastelyn, P.: On the random-cluster model I. Introduction and relation to other models. Physica57, 536–564 (1972)
Fortuin, C., Kastelyn, P., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.22, 89–103 (1971)
Grimmett, G. R.: On the number of clusters in the percolation model. J. Lond. Math. Soc. (2)13, 346–350 (1976)
Grimmett, G. R.: On the differentiability of the number of clusters per vertex in the percolation model. J. Lond. Math. Soc. (2)23, 372–384 (1981)
Grimmett, G. R., Keane, M., Marstrand, J. M.: On the connectedness of a random graph. Math. Proc. Comb. Philos. Soc.96, 151–166 (1984)
Hammersley, J. M.: Percolation processes. Lower bounds for the critical probability. Ann. Math. Statist.28, 790–795 (1957)
Hammersley, J. M.: A Monte Carlo solution of percolation in the cubic crystal. Meth. Comp. Phys.1, 281–298 (1963)
Hankey, A.: Three properties of the infinite cluster in percolation theory. J. Phys. A11, L49-L55 (1978)
Harris, T. E.: A lower bound for the critical probability in a certain percolation process. Proc. Camb. Philos. Soc.56, 13–20 (1960)
Kesten, H.: Percolation theory for mathematicians. Boston: Birkhäuser 1982
Kesten, H.: The incipient infinite cluster in two-dimensional percolation. Theor. Probab. Rel. Fields,73, 369–394 (1986)
Kesten, H.: A scaling relation at criticality for 2D-percolation. In: Percolation theory and ergodic theory of infinite particle systems Kesten, H., (ed.). IMA volumes in mathematics and its applications, Vol.8, Berlin, Heidelberg, New York: Springer (to appear)
Kikuchi, R.: Concept of the long-range order in percolation problems, J. Chem. Phys.53, 2713–2718 (1970)
Kastelyn, P. W., Fortuin, C. M.: Phase transitions in lattice systems with random local properties. J. Phys. Soc. Jpn.26, [Suppl], 11–14 (1969)
Klein, S. T., Shamir, E.: An algorithmic method for studying percolation clusters. Stanford Univ. Dept. of Computer Science, Report no. STAN-CS-82-933 (1982)
Kunz, H., Souillard, B.: Essential singularity in percolation problems and asymptotic behavior of cluster size distribution. J. Stat. Phys.19, 77–106 (1978)
Leath, P. L.: Cluster shape and critical exponents near percolation threshold. Phys. Rev. Lett.36, 921–924 (1976)
Leath, P. L.: Cluster shape and boundary distribution near percolation threshold. Phys. Rev. B14, 5046–5055 (1976)
Lebowitz, J. L.: Coexistence of phases in Ising ferromagnets. J. Stat. Phys.16, 463–476 (1977)
Newman, C. M.: In equalities for γ and related critical exponents in short and long range percolation. In: Percolation theory and ergodic theory of infinite particle systems Kesten, H., (ed.). IMA volumes in mathematics and its applications, Vol8. Berlin, Heidelberg, New York: Springer (to appear)
Newman, C. M.: Some critical exponent inequalities for percolation. J. Stat. Phys.45, 359–368 (1986)
Newman, C. M., Schulman, L. S.: Infinite clusters in percolation models. J. Stat. Phys.26, 613–628 (1981)
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)
Pike, R., Stanley, H. E.: Order propagation near the percolation threshold. J. Phys. A14, L169-L177 (1981)
Rockafellar, T. R.: Convex analysis. Princeton, NJ: Princeton Univ. Press 1970
Ruelle, D.: Statistical mechanics: Rigorous results. New York: W. A. Benjamin 1969
Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor Verw. Geb.56, 229–237 (1981)
Sykes, M. F., Essam, J. W.: Exact critical percolation probabilities for site and bond problems in two dimensions. J. Math. Phys.5, 1117–1127 (1964)
Wierman, J. C.: On critical probabilities in percolation theory. J. Math. Phys.19, 1979–1982 (1978)
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Communicated by M. E. Fisher
Research supported in part by NSF Grant PHY-8605164
Research supported in part by the NSF through a grant to Cornell University
Research supported in part by NSF Grant DMS-8514834
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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, 505–531 (1987). https://doi.org/10.1007/BF01219071
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DOI: https://doi.org/10.1007/BF01219071