×

Mean-field critical behaviour for percolation in high dimensions. (English) Zbl 0698.60100

Summary: The triangle condition for percolation states that \(\sum_{x,y}\tau (0,x)\tau (x,y)\tau (y,0)\) is finite at the critical point, where \(\tau\) (x,y) is the probability that the sites x and y are connected. We use an expansion related to the lace expansion for a self-avoiding walk to prove that the triangle condition is satisfied in two situations: (i) for nearest-neighbour independent bond percolation on the d-dimensional hypercubic lattice, if d is sufficiently large, and (ii) in more than six dimensions for a class of “spread-out” models of independent bond percolation which are believed to be in the same universality class as the nearest-neighbour model. The class of models in (ii) includes the case where the bond occupation probability is constant for bonds of length less than some large number, and is zero otherwise.
In the course of the proof an infrared bound is obtained. The triangle condition is known to imply that various critical exponents take their mean-field (Bethe lattice) values \((\gamma =\beta =1\), \(\delta =\Delta_ t=2\), \(t\geq 2)\) and that the percolation density is continuous at the critical point. We also prove that \(\nu_ 2=1/2\) in (i) and (ii), where \(\nu_ 2\) is the critical exponent for the correlation length.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation

References:

[1] Aizenman, M.: Geometric analysis of {\(\phi\)}4 fields and Ising models, Parts I and II. Commun. Math. Phys.86, 1–48 (1982) · Zbl 0533.58034 · doi:10.1007/BF01205659
[2] 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
[3] Aizenman, M., Fernández, R.: On the critical behaviour of the magnetization in high dimensional Ising models. J. Stat. Phys.44, 393–454 (1986) · Zbl 0629.60106 · doi:10.1007/BF01011304
[4] Aizenman, M., Graham, R.: On the renormalized coupling constant and the susceptibility in {\(\lambda\)}{\(\phi\)} d 4 field theory and the Ising model in four dimensions. Nucl. Phys. B225 [FS9], 261–288 (1983) · doi:10.1016/0550-3213(83)90053-6
[5] 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) · Zbl 0642.60102 · doi:10.1007/BF01219071
[6] Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behaviour in percolation models. J. Stat. Phys.36, 107–143 (1984) · Zbl 0586.60096 · doi:10.1007/BF01015729
[7] Aizenman, M., Simon, B.: Local Ward identities and the decay of correlations in ferromagnets. Commun. Math. Phys.77, 137–143 (1980) · doi:10.1007/BF01982713
[8] Barsky, D.J., Aizenman, M.: Percolation critical exponents under the triangle condition. Preprint (1988) · Zbl 0747.60093
[9] van den Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability, J. Appl. Prob.22, 556–569 (1985) · Zbl 0571.60019 · doi:10.2307/3213860
[10] Broadbent, S.R., Hammersley, J.M.: Percolation processes. I. Crystals and mazes. Proc. Camb. Phil. Soc.53, 629–641 (1957) · Zbl 0091.13901 · doi:10.1017/S0305004100032680
[11] Brydges, D.C., Fröhlich, J., Sokal, A.D.: A new proof of the existence and nontriviality of the continuum {\(\phi\)} 2 4 and {\(\phi\)} 3 4 quantum field theories. Commun. Math. Phys.91, 141–186 (1983) · doi:10.1007/BF01211157
[12] Brydges, D.C., Spencer, T.: Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys.97, 125–148 (1985) · Zbl 0575.60099 · doi:10.1007/BF01206182
[13] Chayes, J.T., Chayes, L.: On the upper critical dimension of Bernoulli percolation. Commun. Math. Phys.113, 27–48 (1987) · Zbl 0627.60100 · doi:10.1007/BF01221395
[14] Essam, J.W.: Percolation Theory. Rep. Prog. Phys.43, 833–912 (1980) · doi:10.1088/0034-4885/43/7/001
[15] Fröhlich, J.: On the triviality of {\(\phi\)} d 4 theories and the approach to the critical point in \(d\mathop > \limits_{( = )} 4\) dimensions. Nucl. Phys. B200 [FS4], 281–296 (1982) · doi:10.1016/0550-3213(82)90088-8
[16] Fröhlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions, and continuous symmetry breaking. Commun. Math. Phys.50, 79–95 (1976) · doi:10.1007/BF01608557
[17] Grimmett, G.: Percolation, Berlin Heidelberg New York: Springer 1989
[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.: Bornes supérieures de la probabilité critique dans un processus de filtration. In: Le Calcul des Probabilités et ses Applications 17–37 CNRS Paris (1959) · Zbl 0096.11502
[20] Hara, T.: Mean field critical behaviour of correlation length for percolation in high dimensions. Preprint (1989)
[21] Hara, T., Slade, G.: On the upper critical dimension of lattice trees and lattice animals. Submitted to J. Stat. Phys. · Zbl 0718.60117
[22] Hara, T., Slade, G.: Unpublished
[23] Kac, M., Uhlenbeck, G.E., Hemmer, P.C.: On the van der Waals theory of the vapor-liquid equilibrium. I. Discussion of a one-dimensional model. J. Math. Phys.4, 216–288 (1963) · Zbl 0938.82517 · doi:10.1063/1.1703946
[24] Kesten, H.: Percolation theory and first passage percolation. Ann. Probab.15, 1231–1271 (1987) · Zbl 0629.60103 · doi:10.1214/aop/1176991975
[25] Lawler, G.: The infinite self-avoiding walk in high dimensions. To appear in Ann. Probab. (1989) · Zbl 0691.60062
[26] Lebowitz, J.L., Penrose, O.: Rigorous treatment of the van der Waals-Maxwell theory of the liquid-vapor transition. J. Math. Phys.7, 98–113 (1966) · Zbl 0938.82520 · doi:10.1063/1.1704821
[27] Menshikov, M.V., Molchanov, S.A., Sidorenko, A.F.: Percolation theory and some applications, Itogi Nauki i Tekhniki (Series of Probability Theory, Mathematical Statistics, Theoretical Cybernetics)24, 53–110 (1986). English translation. J. Soviet Math.42, 1766–1810 (1988) · Zbl 0647.60103
[28] Nguyen, B.G.: Gap exponents for percolation processes with triangle condition. J. Stat. Phys.49, 235–243 (1987) · Zbl 0962.82521 · doi:10.1007/BF01009960
[29] Park, Y.M.: Direct estimates on intersection probabilities of random walks. To appear in J. Stat. Phys. · Zbl 0716.60081
[30] Russo, L.: On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete.56, 229–237 (1981) · Zbl 0457.60084 · doi:10.1007/BF00535742
[31] Slade, G.: The diffusion of self-avoiding random walk in high dimensions. Commun. Math. Phys.110, 661–683 (1987) · Zbl 0628.60073 · doi:10.1007/BF01205555
[32] Slade, G.: The scaling limit of self-avoiding random walk in high dimensions. Ann. Probab.17, 91–107 (1989) · Zbl 0664.60069 · doi:10.1214/aop/1176991496
[33] Slade, G.: Convergence of self-avoiding random walk to Brownian motion in high dimensions. J. Phys. A: Math. Gen.21, L417-L420 (1988) · Zbl 0653.60061 · doi:10.1088/0305-4470/21/7/010
[34] Slade, G.: The lace expansion and the upper critical dimension for percolation, Lectures notes from the A.M.S. Summer Seminar, Blacksburg, June 1989
[35] Sokal, A.D.: A rigorous inequality for the specific heat of an Ising or {\(\phi\)}4 ferromagnet. Phys. Lett.71A, 451–453 (1979)
[36] Sokal, A.D., Thomas, L.E.: Exponential convergence to equilibrium for a class of random walk models. J. Stat. Phys.54, 797–828 (1989) · Zbl 0668.60061 · doi:10.1007/BF01019776
[37] Stauffer, D.: Introduction to percolation theory. Taylor and Francis, London Philadelphia (1985) · Zbl 0990.82530
[38] Tasaki, H.: Hyperscaling inequalities for percolation. Commun. Math. Phys.113, 49–65 (1987) · Zbl 0627.60101 · doi:10.1007/BF01221396
[39] Tasaki, H.: Private communication
[40] Yang, W., Klein, D.: A note on the critical dimension for weakly self avoiding walks. Prob. Th. Rel. Fields79, 99–114 (1988) · Zbl 0631.60076 · doi:10.1007/BF00319107
[41] Ziff, R.M., Stell, G.: Critical behaviour in three-dimensional percolation: Is the percolation threshold a Lifshitz point? Preprint (1988)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.