Abstract
We prove Ornstein–Zernike behaviour in every direction for finite connection functions of the random cluster model on \(\mathbb {Z}^{d},d\ge 3,\) for \(q\ge 1,\) when occupation probabilities of the bonds are close to \(1.\) Moreover, we prove that equi-decay surfaces are locally analytic, strictly convex, with positive Gaussian curvature.
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Bricmont, J., Fröhlich, J.: Statistical mechanical methods in particle structure analysis of lattice field theories II. Scalar Surface Models Commun. Math. Phys. 98, 553–578 (1985)
van den Berg, J., Häggström, O., Kahn, J.: Some conditional correlation inequalities for percolation and related processes. Random Struct. Algorithms 29(4), 417–435 (2006)
Bollobás, B.: Modern Graph Theory. Springer, Berlin (1998)
Braga, G.A., Procacci, A., Sanchis, R.: Ornstein–Zernike behaviour for Bernoulli bond percolation on \(\mathbb{Z}^{d} \)in the supercritical regime. Commun. Pure Appl. Anal. 3(4), 581–606 (2004)
Campanino, M., Chayes, J.T., Chayes, L.: Gaussian fluctuations in the subcritical regime of percolation. Probab. Theory Relat. Fields 88, 269–341 (1991)
Campanino, M., Gianfelice, M.: On the Ornstein–Zernike behaviour for the Bernoulli bond percolation on \(Z^{d}, d\ge 3,\) in the supercitical regime. J. Stat. Phys. 145, 1407–1422 (2011)
Campanino, M., Ioffe, D.: Ornstein–Zernike theory for the Bernoulli bond Percolation on \(\mathbb{Z}^{d}\). Ann. Probab. 30(2), 652–682 (2002)
Campanino, M., Ioffe, D., Louidor, O.: Finite connections for supercritical Bernoulli bond percolation in 2D. Markov Proc. Rel. Fields 16, 225–266 (2010)
Campanino, M., Ioffe, D., Velenik, Y.: Ornstein-Zernike theory for the finite range Ising models above \(T_{c}\). Probab. Theory Relat. Fields 125, 305–349 (2003)
Campanino, M., Ioffe, D., Velenik, Y.: Fluctuation theory of connectivities for subcritical random cluster models. Ann. Probab. 36, 1287–1321 (2008)
Coquille, L., Duminil-Copin, H., Ioffe, D., Velenik, Y.: On the Gibbs states of the non-critical Potts model on \(\mathbb{Z}^{2}\) Probab. Theory Relat. Fields 158, 477–512 (2014)
Edwards, R.G., Sokal, A.D.: Generalization of the Fortuin–Kasteleyn–Swendsen–Wang representation and Monte Carlo algorithm. Phys. Rev. D 38, 2009–2012 (1988)
Fortuin, C., Kasteleyn, P.: On the random-cluster model I. Introduction and relation to other models. Physica 57, 536–564 (1972)
Gallavotti, G.: The phase separation line in the two-dimensional ising model. Commun. Math. Phys. 27, 103–136 (1972)
Georgii, H.-O.: Gibbs Measures and Phase Transition, vol. 9, 2nd edn. De Gruyter Studies in Mathematics (2011)
Grimmett, G.: Random-Cluster Model A Series of Comprehensive Studies in Mathematics, vol. 333. Springer, Berlin (2009)
Kotecký, R., Preiss, D.: Cluster expansion for abstract polymer models. Commun. Math. Phys. 103, 491–498 (1986)
Procacci, A., Scoppola, B.: Analyticity and mixing properties for random cluster model with \(q{\>}0\) on \(\mathbb{Z}^{d}\). J. Stat. Phys. 123, 1285–1310 (2006)
Acknowledgments
We thank R. van den Berg for pointing out the results of reference [2] on which Proposition 4 is based and the referee for useful comments. M. C. and M. G. are partially supported by G.N.A.M.P.A.
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Campanino, M., Gianfelice, M. On the Ornstein–Zernike Behaviour for the Supercritical Random-Cluster Model on \(\mathbb {Z}^{d},d\ge 3\) . J Stat Phys 159, 1456–1476 (2015). https://doi.org/10.1007/s10955-015-1222-0
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DOI: https://doi.org/10.1007/s10955-015-1222-0