Abstract
We consider the plus-phase of the two-dimensional Ising model below the critical temperature. In 1989 Schonmann proved that the projection of this measure onto a one-dimensional line is not a Gibbs measure. After many years of continued research which have revealed further properties of this measure, the question whether or not it is a Gibbs measure in an almost sure sense remains open. In this paper we study the same measure by interpreting it as a temporal process. One of our main results is that the Schonmann projection is almost surely a regular g-measure. That is, it does possess the corresponding one-sided notion of almost Gibbsianness. We further deduce strong one-sided mixing properties which are of independent interest. Our proofs make use of classical coupling techniques and some monotonicity properties which are known to hold for one-sided, but not two-sided conditioning for FKG measures.
References
Aizenman, M.: Translation invariance and instability of phase coexistence in the two-dimensional Ising system. Commun. Math. Phys. 73(1), 83–94 (1980)
Berg, J., Steif, J.E.: On the existence and nonexistence of finitary codings for a class of random fields. Ann. Probab. 27(3), 1501–1522 (1999)
Berghout, S., Fernandez, R., Verbitskiy, E.: On the relation between Gibbs and \(g\)-measures. Ergod. Theory Dynam. Syst. (2018). https://doi.org/10.1017/etds.2018.13
Bissacot, R., Ossami Endo, E., van Enter, A.C.D., Le Ny, A.: Entropic repulsion and lack of the \(g\)-measure property for Dyson models. ArXiv e-prints: arXiv:1705.03156
Chayes, J.T., Chayes, L., Schonmann, R.H.: Exponential decay of connectivities in the two-dimensional Ising model. J. Statist. Phys. 49(3–4), 433–445 (1987)
Dobrushin, R.L.: Description of a random field by means of conditional probabilities and conditions for its regularity. Teor. Verojatnost. i Primenen 13, 201–229 (1968)
Dobrushin, R.L., Shlosman, S.B.: Completely analytical Gibbs fields. Birkhäuser Boston, Boston (1985)
Dobrushin, R.L., Shlosman, S.B.: Gibbsian description of "non-Gibbs" fields. Uspekhi Mat. Nauk 52(2(314)), 45–58 (1997)
Enter, A.C.D., Fernández, R., Sokal, A.D.: Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Stat. Phys. 72(5–6), 879–1167 (1993)
Fernández, R., Maillard, G.: Chains with complete connections and one-dimensional Gibbs measures. Electron. J. Probab. 9(6), 145–176 (2004)
Fernández, R., Pfister, C.-E.: Global specifications and nonquasilocality of projections of Gibbs measures. Ann. Probab. 25(3), 1284–1315 (1997)
Fernández, R., Gallo, S., Maillard, G.: Regular \(g\)-measures are not always Gibbsian. Electron. Commun. Probab. 16, 732–740 (2011)
Friedli, S., Velenik, Y.: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge University Press, Cambridge (2017)
Gallesco, C., Gallo, S., Takahashi, D.Y.: Dynamic uniqueness for stochastic chains with unbounded memory. Stoch. Process. Appl. 128(2), 689–706 (2018)
Gallo, S., Paccaut, F.: On non-regular \(g\)-measures. Nonlinearity 26(3), 763–776 (2013)
Gallo, S., Takahashi, D.Y.: Attractive regular stochastic chains: perfect simulation and phase transition. Ergod. Theory Dynam. Syst. 34(5), 1567–1586 (2014)
Georgii, H.-O.: Gibbs Measures and Phase Transitions, 2nd edn. Walter de Gruyter & Co., Berlin (2011)
Georgii, H.-O., Häggström, O., Maes, C.: The random geometry of equilibrium phases. In: Phase Transitions and Critical Phenomena, Vol. 18, pp. 1–142. Academic Press, San Diego (2001)
Harris, T.E.: On chains of infinite order. Pacif. J. Math. 5, 707–724 (1955)
Higuchi, Y.: On the absence of non-translation invariant Gibbs states for the two-dimensional Ising model. In: Random fields, Colloq. Math. Soc. Jnos Bolyai, 27, Vols. I, II (Esztergom, 1979), pp. 517–534. North-Holland, Amsterdam/New York (1981)
Hulse, P.: Uniqueness and ergodic properties of attractive \(g\)-measures. Ergod. Theory Dynam. Syst. 11(1), 65–77 (1991)
Kalikow, S.: Random Markov processes and uniform martingales. Isr. J. Math. 71(1), 33–54 (1990)
Keane, M.: Strongly mixing \(g\)-measures. Invent. Math. 16(4), 309–324 (1972)
Lanford, O.E., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Commun. Math. Phys. 13, 194–215 (1969)
Liggett, T.M., Steif, J.E.: Stochastic domination: the contact process, Ising models and FKG measures. Ann. Inst. H. Poincaré Probab. Stat. 42(2), 223–243 (2006)
Lörinczi, J.: Quasilocality of projected Gibbs measures through analyticity techniques. Helv. Phys. Acta 68(7–8), 605–626 (1995)
Lörinczi, J., Vande, K.: A note on the projection of Gibbs measures. J. Stat. Phys. 77(3–4), 881–887 (1994)
Maes, C., Vande, K.: Relative energies for non-Gibbsian states. Commun. Math. Phys. 189(2), 277–286 (1997)
Maes, C., Redig, F., Van Moffaert, A.: The restriction of the Ising model to a layer. J. Stat. Phys. 96(1–2), 69–107 (1999)
Maes, C., Redig, F., Van Moffaert, A.: Almost Gibbsian versus weakly Gibbsian measures. Stoch. Process. Appl. 79(1), 1–15 (1999)
Onicescu, O., Mihoc, G.: Sur les chaînes statistiques. C. R. Acad. Sci. Paris 200, 511–512 (1935)
Pfister, C.E., Velenik, Y.: Macroscopic Description of Phase Separation in the 2D Ising Model. World Scientific Publishers, River Edge (1998)
Samson, P.-M.: Concentration of measure inequalities for Markov chains and \(\Phi \)-mixing processes. Ann. Probab. 28(1), 416–461 (2000)
Schonmann, R.H.: Second order large deviation estimates for ferromagnetic systems in the phase coexistence region. Commun. Math. Phys. 112(3), 409–422 (1987)
Schonmann, R.H.: Projections of Gibbs measures may be non-Gibbsian. Commun. Math. Phys. 124(1), 1–7 (1989)
Schonmann, R.H., Shlosman, S.B.: Complete analyticity for \(2\)D Ising completed. Commun. Math. Phys. 170(2), 453–482 (1995)
Warfheimer, M.: Stochastic domination for the Ising and fuzzy Potts models. Electron. J. Probab. 15(58), 1802–1824 (2010)
Acknowledgements
The authors would like to thank Aernout van Enter for stimulating discussions and valuable comments.
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S.A. Bethuelsen acknowledges the DFG, project GA582/7-2.
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Bethuelsen, S.A., Conache, D. One-Sided Continuity Properties for the Schonmann Projection. J Stat Phys 172, 1147–1163 (2018). https://doi.org/10.1007/s10955-018-2092-z
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DOI: https://doi.org/10.1007/s10955-018-2092-z