Summary
We prove the existence of a real valued random field with parameters in thed-dimensional cubic lattice, such that the distribution of the level set of this random field is a Gibbs state for the nearest neighbour ferromagnetic Ising model. Using this, we prove the continuity of the percolation probability with respect to the parameter (β,h) in the uniqueness region except on the critical curve Γ c ={(β,h c (β))}, whereh c(β) is the critical level of the external field above which percolation takes place.
References
Aizenman, M., Kesten, H., Newman, C.: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Commun. Math. Phys.111, 505–532 (1987)
Berg, van den J., Keane, M.: On the continuity of the percolation probability function. Contemp. Math.26, 61–65 (1984)
Burton, R.M., Keane, M.: Density and uniqueness in percolation. Commun. Math. Phys.121, 501–505 (1989)
Gandolfi, A.: Uniqueness of the infinite cluster for stationary Gibbs states. Ann. Probab.17, 1403–1405 (1989)
Georgii, H.O.: Spontaneous magnetization of randomly dilute ferromagnets. J. Stat. Phys.25, 369–396 (1981)
Georgii, H.O.: On the ferromagnetic and the percolative region of random spin systems. Adv. Appl. Probab.16, 732–765 (1984)
Grimmett, G.R.: Percolation. New York Berlin Heidelberg: Springer 1989
Harris, T.E.: A correlation inequality for Markov processes in partially ordered state spaces. Ann. Probab.5, 451–454 (1977)
Holley, R.A.: Remarks on the FKG inequalities. Commun. Math. Phys.36, 227–231 (1974)
Kamae, T., Krengel, U.: Stochastic partial ordering. Ann. Probab.6, 1044–1049 (1978)
Liggett, T.M.: Interacting Particle systems. Berlin Heidelberg New York: Springer 1985
Molchanov, S.A., Stepanov, A.K.: Percolation in random fields I. Teor. Mat. Fiz.55, 246–256 (1983)
Molchanov, S.A., Stepanov, A.K.: Percolation in random fields II. Teor. Mat. Fiz.55, 419–430 (1983)
Stepanov, A.K.: Scaling of percolation of random fields on Z2. Sov. Math., Dokl.37, 749–753 (1988) [English]
Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat.36, 423–439 (1965)
Suzuki, Y.: (private communication)
Suzuki, Y.: Invariant measures for the multitype voter model. (preprint)
Author information
Authors and Affiliations
Additional information
Supported in part by JSPS, BiBoS, Grant in Aid for Cooperative research no. 62303006 and Grant in Aid for Scientific Research no. 63540168
Rights and permissions
About this article
Cite this article
Higuchi, Y. Level set representation for the Gibbs states of the ferromagnetic ising model. Probab. Th. Rel. Fields 90, 203–221 (1991). https://doi.org/10.1007/BF01192162
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01192162
Keywords
- Stochastic Process
- Probability Theory
- Random Field
- External Field
- Mathematical Biology