Abstract
The equality of two critical points — the percolation thresholdp H and the pointp T where the cluster size distribution ceases to decay exponentially — is proven for all translation invariant independent percolation models on homogeneousd-dimensional lattices (d≧1). The analysis is based on a pair of new nonlinear partial differential inequalities for an order parameterM(β,h), which forh=0 reduces to the percolation densityP ∞ — at the bond densityp=1−e −β in the single parameter case. These are: (1)M≦h∂M/∂h+M 2+βM∂M/∂β, and (2) ∂M/∂β≦|J|M∂M/∂h. Inequality (1) is intriguing in that its derivation provides yet another hint of a “ϕ3 structure” in percolation models. Moreover, through the elimination of one of its derivatives, (1) yields a pair of ordinary differential inequalities which provide information on the critical exponents\(\hat \beta\) and δ. One of these resembles an Ising model inequality of Fröhlich and Sokal and yields the mean field bound δ≧2, and the other implies the result of Chayes and Chayes that\(\hat \beta \leqq 1\). An inequality identical to (2) is known for Ising models, where it provides the basis for Newman's universal relation\(\hat \beta (\delta - 1) \geqq 1\) and for certain extrapolation principles, which are now made applicable also to independent percolation. These results apply to both finite and long range models, with or without orientation, and extend to periodic and weakly inhomogeneous systems.
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References
Kesten, H.: The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys.74, 41–59 (1980)
Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 229–237 (1981)
Aizenman, M., Newman, C.M.: Discontinuity of the percolation density in one-dimensional 1/|x−y|2 percolation models. Commun. Math. Phys.107, 611–647 (1986)
Aizenman, M., Chayes, J.T., Chayes, L., Imbrie, J., Newman, C.M.: An intermediate phase with slow decay of correlations in one-dimensional 1/|x−y|2 Ising and Potts models (in preparation)
Aizenman, M.: Contribution in: Statistical physics and dynamical systems (Proceedings Kösheg 1984). Fritz, J., Jaffe, A., Szasz, D. (eds.). Boston: Birkhäuser 1985
Chayes, J.T., Chayes, L.: Critical points and intermediate phases on wedges of ℤd. J. Phys. A (to appear)
Hammersley, J.M.: Percolation processes. Lower bounds for the critical probability. Ann. Math. Statist.28, 790–795 (1957)
Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys.36, 107–143 (1984)
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)
Griffiths, R.B., Hurst, C.A., Sherman, S.: Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys.11, 790–795 (1970)
Newman, C.M.: Shock waves and mean field bounds. Concavity and analyticity of the magnetization at low temperature. Appendix to contribution in Proceedings of the SIAM workshop on multiphase flow, G. Papanicolau (ed.) (to appear)
Aizenman, M., Fernández, R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys.44, 393–454 (1986)
Harris, A.B., Lubensky, T.C., Holcomb, W.K., Dasgupta, C.: Renormalization group approach to percolation problems. Phys. Rev. Lett.35, 327–330 (1975)
Chayes, J.T., Chayes, L.: An inequality for the infinite cluster density in Bernoulli percolation. Phys. Rev. Lett.56, 1619–1622 (1986)
Fröhlich, J., Sokal, A.D.: The random walk representation of classical spin systems and correlation inequalities. III. Nonzero magnetic field (in preparation)
Fernández, R., Fröhlich, J., Sokal, A.D.: Random-walk models and random-walk representations of classical lattice spin systems (in preparation)
Griffiths, R.B.: Correlations in Ising ferromagnets. II. External magnetic fields. J. Math. Phys.8, 484–489 (1967)
Aizenman, M., Kesten, H., Newman, C.M.: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Submitted to Commun. Math. Phys.
van den Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability. J. Appl. Probab.22, 556–569 (1985)
Kesten, H.: Percolation theory for mathematicians. Boston: Birkhäuser 1982
Griffeath, D.: The basic contact process. Stochastic Processes Appl.11, 151–185 (1981)
Aizenman, M., Barsky, D.J., Fernández, R.: The phase transition in a general class of Ising-type models in sharp. Submitted to J. Stat. Phys.
Chayes, J.T., Chayes, L., Newman, C.M.: Bernoulli percolation above threshold: an invasion percolation analysis. Ann. Probab. (to appear)
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Communicated by J. Fröhlich
Research supported in part by the NSF Grant PHY-8605164
Also in the Physics Department
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Aizenman, M., Barsky, D.J. Sharpness of the phase transition in percolation models. Commun.Math. Phys. 108, 489–526 (1987). https://doi.org/10.1007/BF01212322
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DOI: https://doi.org/10.1007/BF01212322