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Moderate deviations for a Curie–Weiss model with dynamical external field∗

Published online by Cambridge University Press:  04 November 2013

Anselm Reichenbachs
Anselm Reichenbachs*
Affiliation:
Ruhr-Universität Bochum, Fakultät für Mathematik, 44780 Bochum, Germany. anselm.reichenbachs@rub.de
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Abstract

In the present paper we prove moderate deviations for a Curie–Weiss model with external magnetic field generated by a dynamical system, as introduced by Dombry and Guillotin-Plantard in [C. Dombry and N. Guillotin-Plantard, Markov Process. Related Fields 15 (2009) 1–30]. The results extend those already obtained for the Curie–Weiss model without external field by Eichelsbacher and Löwe in [P. Eichelsbacher and M. Löwe, Markov Process. Related Fields 10 (2004) 345–366]. The Curie–Weiss model with dynamical external field is related to the so called dynamic ℤ-random walks (see [N. Guillotin-Plantard and R. Schott, Theory and applications, Elsevier B. V., Amsterdam (2006).]). We also prove a moderate deviation result for the dynamic ℤ-random walk, completing the list of limit theorems for this object.

Keywords

Moderate deviationslarge deviationsstatistical mechanicsCurie–Weiss modeldynamic random walksergodic theory
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 725 - 739
Copyright
© EDP Sciences, SMAI, 2013

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References

Costeniuc, M., Ellis, R.S. and Otto, P. Tak-Hun, Multiple critical behavior of probabilistic limit theorems in the neighborhood of a tricritical point. J. Stat. Phys. 127 (2007) 495–552.
Costeniuc, M., Ellis, R.S. and Touchette, H., Complete analysis of phase transitions and ensemble equivalence for the Curie–Weiss–Potts model. J. Math. Phys. 46 (2005) 063301. Google Scholar
A. Dembo and O. Zeitouni, Large deviations techniques and applications Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin 38 (2010). Corrected reprint of the second edition (1998).
Dinwoodie, I.H. and Zabell, S.L., Large deviations for exchangeable random vectors. Ann. Probab. 20 (1992) 1147–1166. Google Scholar
Dombry, C. and Guillotin-Plantard, N., The Curie–Weiss model with dynamical external field. Markov Process. Related Fields 15 (2009) 1–30. Google Scholar
P. Dupuis and R.S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations. Probab. Stat. John Wiley & Sons Inc., New York (1997). A Wiley-Interscience Publication.
Eichelsbacher, P. and Löwe, M., Moderate deviations for a class of mean-field models. Markov Process. Related Fields 10 (2004) 345–366. Google Scholar
Ellis, R.S., Entropy, large deviations, and statistical mechanics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York 271 (1985). Google Scholar
Ellis, R.S. and Newman, C.M., Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44 (1978) 117–139. Google Scholar
Ellis, R.S., Newman, C.M. and Rosen, J.S., Limit theorems for sums of dependent random variables occurring in statistical mechanics II. Conditioning, multiple phases, and metastability. Z. Wahrsch. Verw. Gebiete 51 (1980) 153–169. Google Scholar
Formentin, M., Külske, C. and Reichenbachs, A., Metastates in mean-field models with random external fields generated by Markov chains. J. Stat. Phys. 146 (2012) 314–329. Google Scholar
N. Guillotin-Plantard and R. Schott, Dynamic random walks. Theory and applications. Elsevier B. V., Amsterdam (2006).
Löwe, M. and Meiners, R., Moderate Deviations for Random Field Curie–Weiss Models. J. Stat. Phys. 149 (2012) 701–721. Google Scholar
K. Petersen, Ergodic Theory, vol. 2 of Adv. Math. Cambridge University Press, Cambridge (1983).