Hostname: page-component-6bf8c574d5-k2jvg Total loading time: 0 Render date: 2025-03-03T16:56:00.719Z Has data issue: false hasContentIssue false
  • English
  • Français

Phase transitions in long-range Ising models and an optimal condition for factors of $g$-measures

Published online by Cambridge University Press:  07 September 2017

ANDERS JOHANSSON ,
ANDERS ÖBERG  and
MARK POLLICOTT
ANDERS JOHANSSON
Affiliation:
Department of Mathematics, University of Gävle, 801 76 Gävle, Sweden email [email protected]
ANDERS ÖBERG
Affiliation:
Department of Mathematics, Uppsala University, PO Box 480, 751 06 Uppsala, Sweden email [email protected]
MARK POLLICOTT
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We weaken the assumption of summable variations in a paper by Verbitskiy [On factors of $g$-measures. Indag. Math. (N.S.)22 (2011), 315–329] to a weaker condition, Berbee’s condition, in order for a one-block factor (a single-site renormalization) of the full shift space on finitely many symbols to have a $g$-measure with a continuous $g$-function. But we also prove by means of a counterexample that this condition is (within constants) optimal. The counterexample is based on the second of our main results, where we prove that there is a critical inverse temperature in a one-sided long-range Ising model which is at most eight times the critical inverse temperature for the (two-sided) Ising model with long-range interactions.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 5 , May 2019 , pp. 1317 - 1330
Copyright
© Cambridge University Press, 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aizenman, M., Chayes, J., Chayes, L. and Newman, C.. Discontinuity of the magnetization in the one-dimensional 1/|x - y|2 Ising and Potts models. J. Stat. Phys. 50 (1988), 1–40.Google Scholar
Berbee, H.. Chains with infinite connections: uniqueness and Markov representation. Probab. Theory Related Fields 76 (1987), 243–253.Google Scholar
Berbee, H.. Uniqueness of Gibbs measures and absorption probabilities. Ann. Probab. 17(4) (1989), 1416–1431.Google Scholar
Berger, N., Hoffman, C. and Sidoravicius, V.. Nonuniqueness for specifications in $\ell ^{2+\unicode[STIX]{x1D716}}$ . Ergod. Th. & Dynam. Sys. (2017), to appear.Google Scholar
Cioletti, L. and Lopes, A.. Ruelle operator for continuous potentials and DLR-Gibbs measures. Preprint, 2016, arXiv:1608.03881v1.Google Scholar
Doeblin, W. and Fortet, R.. Sur les châines á liaisons complètes. Bull. Soc. Math. France 65 (1937), 132–148.Google Scholar
van Enter, A. C. D., Fernandez, R. and 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) (1993), 879–1167.Google Scholar
Fan, A. H. and Pollicott, M.. Non-homogeneous equilibrium states and convergence speeds of averaging operators. Math. Proc. Cambridge Philos. Soc. 129(1) (2000), 99–115.Google Scholar
Frölich, J. and Spencer, T.. The phase transition in the one-dimensional Ising model with 1/r 2 interaction energy. Comm. Math. Phys. 4(1) (1982), 87–101.Google Scholar
Johansson, A. and Öberg, A.. Square summability of variations of g-functions and uniqueness of g-measures. Math. Res. Lett. 10(5–6) (2003), 587–601.Google Scholar
Johansson, A., Öberg, A. and Pollicott, M.. Countable state shifts and uniqueness of g-measures. Amer. J. Math. 129 (2007), 1501–1511.Google Scholar
Johansson, A., Öberg, A. and Pollicott, M.. Unique Bernoulli g-measures. J. Eur. Math. Soc. (JEMS) 14 (2012), 1599–1615.Google Scholar
Keane, M.. Strongly mixing g-measures. Invent. Math. 16 (1972), 309–324.Google Scholar
Redig, F. and Wang, F.. Transformations of one-dimensional Gibbs measures with infinite range interaction. Markov Process. Related Fields 16(4) (2010), 737–752.Google Scholar
Sinai, Ya. G.. Gibbs measures in ergodic theory. Russian Math. Surveys 27(4) (1972), 21–69.Google Scholar
Verbitskiy, E.. On factors of g-measures. Indag. Math. (N.S.) 22 (2011), 315–329.Google Scholar
Walters, P.. Convergence of the Ruelle operator. Trans. Amer. Math. Soc. 353(1) (2000), 327–347.Google Scholar