zbMATH — the first resource for mathematics

Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. (English) Zbl 0642.60102
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 $$\alpha$$ in terms of the mean cluster size exponent $$\gamma$$ and the critical cluster size distribution exponent $$\delta$$ ; e.g. $$1+\alpha \leq \gamma (\delta /2-1)/(\delta -1)$$.

MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation
Full Text:
References:
 [1] Aizenman, M., Barsky, D. J.: Sharpness of the phase transition in percolation models, Commun. Math. Phys.108, 489-526 (1987). · Zbl 0618.60098 · doi:10.1007/BF01212322 [2] Aizenman, M., Barsky, D. J.: in preparation. See also Barsky, D. J. Rutgers University Ph.D. thesis (1987). · Zbl 0618.60098 [3] Aizenman, M., Chayes, J. T., Chayes, L., Fr?hlich, 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) · Zbl 0529.60099 · doi:10.1007/BF01206313 [4] Aizenman, M., Newman, C. M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys.36, 107-143 (1984) · Zbl 0586.60096 · doi:10.1007/BF01015729 [5] 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) · Zbl 0613.60097 · doi:10.1007/BF01205489 [6] van den Berg, J., Keane, M.: On the continuity of the percolation probability function, Contemp. Math.26, 61-65 (1984) · Zbl 0541.60099 [7] Bricmont, J., Lebowitz, J. L.: On the continuity of the magnetization and energy in Ising ferromagnets. J. Stat. Phys.42, 861-869 (1986) · doi:10.1007/BF01010449 [8] Choquet, G.: Topology. New York: Academic Press 1966 [9] 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) [10] Chayes, J. T., Chayes, L., Newman, C. M., The stochastic geometry of invasion percolation. Commun. Math. Phys.101, 383-407 (1985) · Zbl 0596.60096 · doi:10.1007/BF01216096 [11] Durrett, R., Nguyen, B.: Thermodynamic inequalities for percolation. Commun. Math. Phys.99, 253-269 (1985) · doi:10.1007/BF01212282 [12] Fisher, M. E.: Critical probabilities for cluster size and percolation problems. J. Math. Phys.2, 620-627 (1961) · Zbl 0105.43602 · doi:10.1063/1.1703746 [13] Fortuin, C., Kastelyn, P.: On the random-cluster model I. Introduction and relation to other models. Physica57, 536-564 (1972) · doi:10.1016/0031-8914(72)90045-6 [14] Fortuin, C., Kastelyn, P., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.22, 89-103 (1971) · Zbl 0346.06011 · doi:10.1007/BF01651330 [15] Grimmett, G. R.: On the number of clusters in the percolation model. J. Lond. Math. Soc. (2)13, 346-350 (1976) · Zbl 0338.60034 · doi:10.1112/jlms/s2-13.2.346 [16] 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) · Zbl 0497.60010 · doi:10.1112/jlms/s2-23.2.372 [17] Grimmett, G. R., Keane, M., Marstrand, J. M.: On the connectedness of a random graph. Math. Proc. Comb. Philos. Soc.96, 151-166 (1984) · Zbl 0543.60016 · doi:10.1017/S0305004100062034 [18] Hammersley, J. M.: Percolation processes. Lower bounds for the critical probability. Ann. Math. Statist.28, 790-795 (1957) · Zbl 0091.13903 · doi:10.1214/aoms/1177706894 [19] Hammersley, J. M.: A Monte Carlo solution of percolation in the cubic crystal. Meth. Comp. Phys.1, 281-298 (1963) [20] Hankey, A.: Three properties of the infinite cluster in percolation theory. J. Phys. A11, L49-L55 (1978) [21] Harris, T. E.: A lower bound for the critical probability in a certain percolation process. Proc. Camb. Philos. Soc.56, 13-20 (1960) · Zbl 0122.36403 · doi:10.1017/S0305004100034241 [22] Kesten, H.: Percolation theory for mathematicians. Boston: Birkh?user 1982 · Zbl 0522.60097 [23] Kesten, H.: The incipient infinite cluster in two-dimensional percolation. Theor. Probab. Rel. Fields,73, 369-394 (1986) · Zbl 0597.60099 · doi:10.1007/BF00776239 [24] 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) [25] Kikuchi, R.: Concept of the long-range order in percolation problems, J. Chem. Phys.53, 2713-2718 (1970) · doi:10.1063/1.1674394 [26] Kastelyn, P. W., Fortuin, C. M.: Phase transitions in lattice systems with random local properties. J. Phys. Soc. Jpn.26, [Suppl], 11-14 (1969) [27] 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) [28] Kunz, H., Souillard, B.: Essential singularity in percolation problems and asymptotic behavior of cluster size distribution. J. Stat. Phys.19, 77-106 (1978) · doi:10.1007/BF01020335 [29] Leath, P. L.: Cluster shape and critical exponents near percolation threshold. Phys. Rev. Lett.36, 921-924 (1976) · doi:10.1103/PhysRevLett.36.921 [30] Leath, P. L.: Cluster shape and boundary distribution near percolation threshold. Phys. Rev. B14, 5046-5055 (1976) [31] Lebowitz, J. L.: Coexistence of phases in Ising ferromagnets. J. Stat. Phys.16, 463-476 (1977) · doi:10.1007/BF01152284 [32] Newman, C. M.: In equalities for ? 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) [33] Newman, C. M.: Some critical exponent inequalities for percolation. J. Stat. Phys.45, 359-368 (1986) · doi:10.1007/BF01021076 [34] Newman, C. M., Schulman, L. S.: Infinite clusters in percolation models. J. Stat. Phys.26, 613-628 (1981) · Zbl 0509.60095 · doi:10.1007/BF01011437 [35] 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) · Zbl 0604.60097 · doi:10.1007/BF01211064 [36] Pike, R., Stanley, H. E.: Order propagation near the percolation threshold. J. Phys. A14, L169-L177 (1981) [37] Rockafellar, T. R.: Convex analysis. Princeton, NJ: Princeton Univ. Press 1970 · Zbl 0193.18401 [38] Ruelle, D.: Statistical mechanics: Rigorous results. New York: W. A. Benjamin 1969 · Zbl 0177.57301 [39] Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor Verw. Geb.56, 229-237 (1981) · Zbl 0457.60084 · doi:10.1007/BF00535742 [40] 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) · doi:10.1063/1.1704215 [41] Wierman, J. C.: On critical probabilities in percolation theory. J. Math. Phys.19, 1979-1982 (1978) · Zbl 0416.60099 · doi:10.1063/1.523894
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.