Complete analyticity for 2D Ising completed
Tóm tắt
We study the behavior of the two-dimensional nearest neighbor ferromagnetic Ising model under an external magnetic fieldh. We extend to every subcritical value of the temperature a result previously proven by Martirosyan at low enough temperature, and which roughly states that for finite systems with — boundary conditions under a positive external field, the boundary effect dominates in the bulk if the linear size of the system is of orderB/h withB small enough, while ifB is large enough, then the external field dominates in the bulk. As a consequence we are able to complete the proof that “complete analyticity for nice sets” holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model. The main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field, and recently extended to all temperatures below the critical one by Ioffe.
Tài liệu tham khảo
[AR] Abraham, D.B., Reed, P.: Diagonal interface in the two-dimensional Ising ferromagnet. J. Phys. A: Math. Gen.10, L121-L123 (1977)
[CCS] Chayes, J., Chayes, L., Schonmann, R.H.: Exponential decay of connectivities in the two-dimensional Ising model. J. Stat. Phys.49, 433–445 (1987)
[DKS] Dobrushin, R.L., Kotecký, R., Shlosman, S.: Wulff construction: A global shape from local interaction. AMS translation series, 1992
[DS1] Dobrushin, R.L., Shlosman, S.: Constructive criterion for the uniqueness of Gibbs fields. In: Statistical Physics and Dynamical Systems, Fritz, J., Jaffe, A., Szász, D., ed., Basel-Boston: Birkhauser, pp. 347–370
[DS2] Dobrushin, R.L., Shlosman, S.: Completely analytical Gibbs fields. In: Statistical Physics and Dynamical Systems, Fritz, J., Jaffe, A., Szász, D., ed., Basel, Boston: Birkhauser, pp. 371–403
[DS3] Dobrushin, R.L., Shlosman, S.: Completely analytical interactions. Constructive description. J. Stat. Phys.46, 983–1014 (1987)
[FO] Föllmer, H., Orey, S.: Large deviations for the empirical field of a Gibbs measure. Ann. Prob.16, 961–977 (1988)
[Hi] Higuchi, Y.: Coexistence of infinite (*)-clusters II:- Ising percolation in two dimensions. Prob. Theory and Related Fields97, 1–34 (1993)
[lofl] Ioffe, D.: Large deviations for the 2D Ising model: A lower bound without cluster expansions. J. Stat. Phys.74, 411–432 (1994)
[lof2] Ioffe, D.: Exact large deviation bounds up toT c for the Ising model in two dimensions. Preprint
[Lan] Lanford, O.E.: Entropy and equilibrium states in classical statistical mechanics. Lecture Notes in Physics, vol.20, Berlin-Heidelberg-New York: Springer, 1973, pp. 1–113
[Lig] Liggett, T.: Interacting Particle Systems, Berlin, Heidelberg. New York: Springer
[LY] Lu, S.L., Yau, H.T.: Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Commun. Math. Phys.156, 399–433 (1993)
[Mar] Martirosyan, D.G.: Theorems on strips in the classical Ising ferromagnetic model. Sov. J. Contemp. Math. Anal.22, 59–83 (1987)
[MO1] Martinelli, F., Olivieri, E.: Approach to equilibrium of Glauber dynamics in the one phase region I. The attractive case. Commun. Math. Phys.161, 447–486 (1994)
[MO2] Martinelli, F., Olivieri, E.: Approach to equilibrium of Glauber dynamics in the one phase region II. The general case. Commun. Math. Phys.161, 487–514 (1994)
[MOS] Martinelli, F., Olivieri, E., Schonmann, R.H.: For 2-D lattice spin systems weak mixing implies strong mixing. Commun. Math. Phys.,165, 33–47 (1994)
[Oll] Olla, S.: Large deviations for Gibbs random fields, Prob. Th. Rel. Fields77, 343–357 (1988)
[Pfi] Pfister, C.E.: Large deviations and phase separation in the two-dimensional Ising model. Helv. Phys. Acta64, 953–1054 (1991)
[Sch1] Schonmann, R.H.: Exponential convergence under mixing. Prob. Th. Rel. Fields81, 235–238 (1989)
[Sch2] Schonmann, R.H.: Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region. Commun. Math. Phys.161, 1–49 (1994)
[SS] Schonmann, R.H., Shlosman, S.B.: Constrained variational problem with applications to the Ising model. In preparation
[Sh1] Shlosman, S.B.: Uniqueness and half-space nonuniqueness of Gibbs states in Czech model. Theor. Math. Phys.66, 284–293 (1986)
[SZ] Stroock, D.W., Zegarlinski, B.: The logarithmic Sobolev inequality for Discrete spin systems on a lattice. Commun. Math. Phys.149, 175–194 (1992)