Abstract
We prove that the critical value β c of a ferromagnetic Potts model is astrictly decreasing function of the strengths of interaction of the process. This is achieved in the (more) general context of the random-cluster representation of Fortuin and Kasteleyn, by deriving and utilizing a formula which generalizes the technique known in percolation theory as Russo's formula. As a byproduct of the method, we present a general argument for showing that, at any given point on the critical surface of a multiparameter process, the values of a certain critical exponent do not depend on the direction of approach of that point. Our results apply to all random-cluster processes satisfying the FKG inequality.
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Aizenman, M.: Geometric analysis of φ4 fields and Ising models. Commun. Math. Phys.86, 1–48 (1982)
Aizenman, M., Chayes, J.T., Chayes, L., and Newman, C.M.: Discontinuity of the magnetization in one-dimensional 1/|x−y|2 Ising and Potts models. J. Stat. Phys.50, 1–40 (1988)
Aizenman, M., Fernández, R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys.44, 393–454 (1986)
Aizenman, M., Grimmett, G. R.: Strict monotonicity for critical points in percolation and ferromagnetic models. J. Stat. Phys.63, 817–835 (1991)
Barlow, R.N., Proschan, F.: Mathematical Theory of Reliability. New York: Wiley 1965
Doob, J. L.: Stochastic Processes. New York: Wiley 1953
Edwards, R.G., Sokal, A.D.: Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D38, 2009–2012 (1988)
Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.22, 89–103 (1971)
Grimmett, G.R.: Percolation. Berlin, Heidelberg, New York: Springer 1989
Grimmett, G.R.: Potts models and random-cluster processes with many-body interactions. Preprint (1992)
Grimmett, G.R.: The random-cluster model. Preprint (1993)
Harris, T.E.: Contact interactions on a lattice. Ann. Probab.2, 969–988 (1974)
Harris, T. E.: Additive set-valued Markov processes and graphical methods. Ann. Probab.6, 355–378 (1978)
Holley, R.: Remarks on the FKG inequalities. Commun. Math. Phys.36, 227–231 (1974)
Menshikov, M.: Quantitative estimates and rigorous inequalities for critical points of a graph and its subgraphs. Theory of Probab. and its Appl.32, 544–547 (1987)
Preston, C.J.: Gibbs States on Countable Graphs. Cambridge: Cambridge University Press 1974
Wierman, J.C.: Equality of directional critical exponents in multiparameter percolation models. In preparation (1992)
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Communicated by M. Aizenman
G.R.G. acknowledges support from Cornell University, and also partial support by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University. H.K. was supported in part by the N.S.F. through a grant to Cornell University
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Bezuidenhout, C.E., Grimmett, G.R. & Kesten, H. Strict inequality for critical values of Potts models and random-cluster processes. Commun.Math. Phys. 158, 1–16 (1993). https://doi.org/10.1007/BF02097229
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DOI: https://doi.org/10.1007/BF02097229