Abstract
The triangle condition for percolation states that\(\sum\limits_{x,y} {\tau (0,x)\tau (0,y) \cdot \tau (y,0)} \) is finite at the critical point, where τ(x, y) is the probability that the sitesx andy 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 thed-dimensional hypercubic lattice, ifd 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\((\gamma = \beta = 1,\delta = \Delta _t = 2, t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 2)\) and that the percolation density is continuous at the critical point. We also prove thatv 2 in (i) and (ii), wherev 2 is the critical exponent for the correlation length.
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Aizenman, M.: Geometric analysis of φ4 fields and Ising models, Parts I and II. Commun. Math. Phys.86, 1–48 (1982)
Aizenman, M., Barsky, D.J.: Sharpness of the phase transition in percolation models. Commun. Math. Phys.108, 489–526 (1987)
Aizenman, M., Fernández, R.: On the critical behaviour of the magnetization in high dimensional Ising models. J. Stat. Phys.44, 393–454 (1986)
Aizenman, M., Graham, R.: On the renormalized coupling constant and the susceptibility in λφ 4d field theory and the Ising model in four dimensions. Nucl. Phys. B225 [FS9], 261–288 (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, 505–531 (1987)
Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behaviour in percolation models. J. Stat. Phys.36, 107–143 (1984)
Aizenman, M., Simon, B.: Local Ward identities and the decay of correlations in ferromagnets. Commun. Math. Phys.77, 137–143 (1980)
Barsky, D.J., Aizenman, M.: Percolation critical exponents under the triangle condition. Preprint (1988)
van den Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability, J. Appl. Prob.22, 556–569 (1985)
Broadbent, S.R., Hammersley, J.M.: Percolation processes. I. Crystals and mazes. Proc. Camb. Phil. Soc.53, 629–641 (1957)
Brydges, D.C., Fröhlich, J., Sokal, A.D.: A new proof of the existence and nontriviality of the continuum φ 42 and φ 43 quantum field theories. Commun. Math. Phys.91, 141–186 (1983)
Brydges, D.C., Spencer, T.: Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys.97, 125–148 (1985)
Chayes, J.T., Chayes, L.: On the upper critical dimension of Bernoulli percolation. Commun. Math. Phys.113, 27–48 (1987)
Essam, J.W.: Percolation Theory. Rep. Prog. Phys.43, 833–912 (1980)
Fröhlich, J.: On the triviality of φ 4d theories and the approach to the critical point in\(d\mathop > \limits_{( = )} 4\) dimensions. Nucl. Phys. B200 [FS4], 281–296 (1982)
Fröhlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions, and continuous symmetry breaking. Commun. Math. Phys.50, 79–95 (1976)
Grimmett, G.: Percolation, Berlin Heidelberg New York: Springer 1989
Hammersley, J.M.: Percolation processes. Lower bounds for the critical probability. Ann. Math. Statist.28, 790–795 (1957)
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)
Hara, T.: Mean field critical behaviour of correlation length for percolation in high dimensions. Preprint (1989)
Hara, T., Slade, G.: On the upper critical dimension of lattice trees and lattice animals. Submitted to J. Stat. Phys.
Hara, T., Slade, G.: Unpublished
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)
Kesten, H.: Percolation theory and first passage percolation. Ann. Probab.15, 1231–1271 (1987)
Lawler, G.: The infinite self-avoiding walk in high dimensions. To appear in Ann. Probab. (1989)
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)
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)
Nguyen, B.G.: Gap exponents for percolation processes with triangle condition. J. Stat. Phys.49, 235–243 (1987)
Park, Y.M.: Direct estimates on intersection probabilities of random walks. To appear in J. Stat. Phys.
Russo, L.: On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete.56, 229–237 (1981)
Slade, G.: The diffusion of self-avoiding random walk in high dimensions. Commun. Math. Phys.110, 661–683 (1987)
Slade, G.: The scaling limit of self-avoiding random walk in high dimensions. Ann. Probab.17, 91–107 (1989)
Slade, G.: Convergence of self-avoiding random walk to Brownian motion in high dimensions. J. Phys. A: Math. Gen.21, L417-L420 (1988)
Slade, G.: The lace expansion and the upper critical dimension for percolation, Lectures notes from the A.M.S. Summer Seminar, Blacksburg, June 1989
Sokal, A.D.: A rigorous inequality for the specific heat of an Ising or φ4 ferromagnet. Phys. Lett.71A, 451–453 (1979)
Sokal, A.D., Thomas, L.E.: Exponential convergence to equilibrium for a class of random walk models. J. Stat. Phys.54, 797–828 (1989)
Stauffer, D.: Introduction to percolation theory. Taylor and Francis, London Philadelphia (1985)
Tasaki, H.: Hyperscaling inequalities for percolation. Commun. Math. Phys.113, 49–65 (1987)
Tasaki, H.: Private communication
Yang, W., Klein, D.: A note on the critical dimension for weakly self avoiding walks. Prob. Th. Rel. Fields79, 99–114 (1988)
Ziff, R.M., Stell, G.: Critical behaviour in three-dimensional percolation: Is the percolation threshold a Lifshitz point? Preprint (1988)
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Communicated by M. Aizenman
Supported by the Nishina Memorial Foundation and NSF grant PHY-8896163. Address after September 1, 1989: Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA.
Supported by NSERC grant A9351.
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Hara, T., Slade, G. Mean-field critical behaviour for percolation in high dimensions. Commun.Math. Phys. 128, 333–391 (1990). https://doi.org/10.1007/BF02108785
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DOI: https://doi.org/10.1007/BF02108785