Abstract
We consider the Fortuin-Kasteleyn cluster model which is related to the percolation model and the Potts model. We give the asymptotic behavior of the critical probability and the percolation probability as the dimension d tends to infinity, in the case where Q (which corresponds to the number of colors in the Potts model) lies between 1 and 2.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aizenman, M, Chayes, J.T., Chayes, L. and Newman, C.M., Discontinuity of the magnetization in one-dimensional l/|x — y| 2 Ising and Potts models, J. Stat. Phys. 50 (1988), pp. 1–40.
Aizenman, M., Kesten, H. and Newman, C.M., Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation, Comm. Math. Phys. 111 (1987), pp. 505–531.
van den Berg, J. and Burton, R., FKG and equivalent conditions for binary random variables, preprint (1987).
Burton, R.M. and Keane, M., Density and uniqueness in percolation, Comm. Math. Phys. 121 (1989), pp. 501–505.
Bricmont, J., Kesten, H., Lebowitz, J. and Schonmann, R.H., A note on the large dimensional Ising model, Comm. Math. Phys. 122 (1989), pp. 597–607.
Edwards, R. and Sokal, A., Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithms, Phys. Rev. D 38 (1988), pp. 2009–2012.
Fisher, M.E., Critical temperatures of anisotropic Ising lattices. III. General upper bounds, Phys. Rev. 162 (1967), pp. 480–485.
Fortuin, C.M., On the random cluster model III. The simple random cluster model, Physica 59 (1972), pp. 545–570.
Gandolfi, A. Keane, M.S. and Newman, C.M., Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses, preprint, Delft University of Technology (1989).
Gandolfi, A., Grimmett, G. and Russo, L., On the uniqueness of the infinite cluster in the percolation model, Comm. Math. Phys. 114 (1988), pp. 549–552.
Harris, T.E., A lower bound for the critical probability in a certain percolation process, Proc. Cambr. Phil. Soc. 56 (1960), pp. 13–20.
Holley, R., Remarks on the FKG inequalities, Comm. Math. Phys. 36, pp. 227–231.
Hara, T. and Slade, G., Mean-field critical phenomena for percolation in high dimensions, Comm. Math. Phys. 128 (1990), pp. 333–391.
Kemperman, J.H.B., On the FKG-inequality for measures on a partially ordered space, Proc. Kon. Ned. Akad. van Wetensch., Ser. A 80 (1977), pp. 313–331.
Kesten, H., Asymptotics in high dimension for percolation, in “Disorder in Physical Systems, A Volume in Honor of J.M. Hammersley,” eds. G. R. Grimmett and D. J. A. Welsh, Oxford University Press, 1990, pp. 219–240.
Kesten, H. and Schonmann, R., Behavior in large dimensions of the Potts and Heisenberg models, Rev. Math. Phys. 1 (1990), pp. 147–182.
Schonmann, R.H. and Vares, M.E., The survival of the large dimensional basic contact process, Prob. Th. Rel. Fields 72 (1986), pp. 387–393.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer Science+Business Media New York
About this chapter
Cite this chapter
Kesten, H. (1991). Asymptotics in High Dimensions For the Fortuin-Kasteleyn Random Cluster Model. In: Alexander, K.S., Watkins, J.C. (eds) Spatial Stochastic Processes. Progress in Probability, vol 19. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0451-0_4
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0451-0_4
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6766-9
Online ISBN: 978-1-4612-0451-0
eBook Packages: Springer Book Archive