Summary
Extending the method of [27], we prove that the corrlation length ξ of independent bond percolation models exhibits mean-field type critical behaviour (i.e. ξ(p∼(p c −p)−1/2 asp↗p c ) in two situations: i) for nearest-neighbour independent bond percolation models on ad-dimensional hypercubic lattice ℤd, withd sufficiently large, and ii) for a class of “spread-out” independent bond percolation models, which are believed to belong to the same universality class as the nearest-neighbour model, in more than six dimensions. The proof is based on, and extends, a method developed in [27], where it was used to prove the triangle condition and hence mean-field behaviour of the critical exponents γ, β, δ, Δ and ν2 for the above two cases.
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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.: Rigorous studies of critical behaviour, II. In: Jaffe, A., Fritz, J., Szász, D. (eds.) Statistical physics and dynamical systems: rigorous results. Boston, Basel, Stuttgart: Birkhäuser 1985/Köszeg 1984
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., 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)
Alexander, K.S., Chayes, J.T., Chayes, L.: The Wulff construction and asymptotics of the finite cluster distribution for two dimensional Bernoulli percolation. (preprint 1989)
Barsky, D.J., Aizenman, M.: Percolation critical exponents under the triangle condition. Ann. Probab. (to appear)
Berg, J. van den, Kesten, H.: Inequalities with applications to percolation and reliability. J. Appl. Prob.22, 556–569 (1985)
Bovier, A., Fröhlich, J., Glaus, U.: Branched polymers and dimensional reduction. In: Osterwalder, K., Stora, R. (eds.) Critical Phenomena, Random Systems, Gauge Theories. Amsterdam, New York, Oxford, Tokyo: North-Holland 1986, Les Houches 1984
Bricmont, J., Kesten, H., Lebowitz, J.L., Schonmann, R.H.: A note on the Ising models in high dimensions. Commun. Math. Phys.122, 597–607 (1989)
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)
Campanino, M., Chayes, J.T., Chayes, L.: Gaussian fluctuations of connectivities in the subcritical regime of percolation. (preprint 1988)
Chayes, J.T., Chayes, L.: Percolation and random media. In: Osterwalder, W., Stora, R. (eds.) Critical Phenomena, Random Systems, Gauge Theories. Amsterdam, New York, Oxford, Tokyo: North-Holland 1986, Les Houches 1984
Chayes, J.T., Chayes, L.: On the upper critical dimension of Bernoulli percolation. Commun. Math. Phys.113, 27–48 (1987)
Fortuin, G., Kastelyn, P., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.22, 89–103 (1971)
Fröhlich, J.: On the triviality of ϕ 4 d theories and the approach to the critical point ind(≧) 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)
Gawędzki, K., Kupiainen, A.: Massless lattice ϕ 44 theory: rigorous control of a renormalizable asymptotically free model. Commun. Math. Phys.99, 199–252 (1985)
Grimmett, G.R.: Percolation, Berlin Heidelberg New York: Springer 1988
Hammersley, J.M.: Percolation processes, II. Connective constants. Proc. Camb. Phil. Soc.53, 642–645 (1957)
Hammersley, J.M.: Percolation processes, IV. Lower bounds for critical probabilities. Ann. Math. Statist.28, 790–794 (1957)
Hammersley, J.M.: Bornes supérieures de la probabilité critique dans un processus de filtration. In: Le calcul des probabilités et ses applications, p. 17–37, (1959). CNRS, Paris
Hara, T.: A rigorous control of logarithmic corrections in four dimensional ϕ4 spin systems. I. Trajectory of effective Hamiltonians. J. Stat. Phys.47, 57–98 (1987)
Hara, T., Slade, G.: Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys.128, 333–391 (1990). Announced in: The triangle condition for percolation. Bull. Ann. Math. Soc., New Ser.21, 269–273 (1989)
Hara, T., Slade, G.: On the upper critical dimension of lattice trees and lattice animals. J. Stat. Phys.59, 1469–1510 (1990)
Hara, T., Tasaki, H.: A rigorous control of logarithmic corrections in four dimensional ϕ4 spin systems. II. Critical behaviour of susceptibility and correlation length. J. Stat. Phys.47, 99–121 (1987)
Harris, T.E.: A lower bound for the critical probability in certain percolation processes. Proc. Camb. Phil. Soc.56, 13–20 (1960)
Lubensky, T.C., Isaacson, J.: Statistics of lattice animals and dilute branched polymers. Phys. Rev.A20, 2130–2146 (1979)
Menshikov, M.V.: Coincidence of critical points in percolation problems. Sov. Math. Dokl.33, 856–859 (1986)
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. Sov. Math.42, 1766–1810 (1988)
Nguyen, B.G.: Correlation length and its critical exponent for percolation processes. J. Stat. Phys.46, 517–523 (1987)
Nguyen, B.G.: Gap exponents for percolation processes with triangle condition. J. Stat. Phys.49, 235–243 (1987)
Simon, B.: Correlation inequalities and the decay of correlations in ferromagnets. Commun. Math. Phys.77, 111–126 (1980)
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)
Sokal, A.D.: A rigorous inequality for the specific heat of an Ising of ϕ4 ferromagnet. Phys. Lett.71 A, 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)
Tasaki, H.: Hyperscaling inequalities for percolation. Commun. Math. Phys.113, 49–65 (1987)
Ziff, R.M., Stell, G.: Critical behaviour in three dimensional percolation: is the percolation threshold a Lifshitz point? (preprint)
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Hara, T. Mean-field critical behaviour for correlation length for percolation in high dimensions. Probab. Th. Rel. Fields 86, 337–385 (1990). https://doi.org/10.1007/BF01208256
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DOI: https://doi.org/10.1007/BF01208256