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.
Mots clés : moderate deviations, large deviations, statistical mechanics, Curie-Weiss model, dynamic random walks, ergodic theory
@article{PS_2013__17__725_0,
author = {Reichenbachs, Anselm},
title = {Moderate deviations for a {Curie-Weiss} model with dynamical external field},
journal = {ESAIM: Probability and Statistics},
pages = {725--739},
publisher = {EDP-Sciences},
volume = {17},
year = {2013},
doi = {10.1051/ps/2012019},
mrnumber = {3126159},
zbl = {1290.60105},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2012019/}
}
TY - JOUR AU - Reichenbachs, Anselm TI - Moderate deviations for a Curie-Weiss model with dynamical external field JO - ESAIM: Probability and Statistics PY - 2013 SP - 725 EP - 739 VL - 17 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2012019/ DO - 10.1051/ps/2012019 LA - en ID - PS_2013__17__725_0 ER -
%0 Journal Article %A Reichenbachs, Anselm %T Moderate deviations for a Curie-Weiss model with dynamical external field %J ESAIM: Probability and Statistics %D 2013 %P 725-739 %V 17 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2012019/ %R 10.1051/ps/2012019 %G en %F PS_2013__17__725_0
Reichenbachs, Anselm. Moderate deviations for a Curie-Weiss model with dynamical external field. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 725-739. doi : 10.1051/ps/2012019. http://www.numdam.org/articles/10.1051/ps/2012019/
[1] , and , Multiple critical behavior of probabilistic limit theorems in the neighborhood of a tricritical point. J. Stat. Phys. 127 (2007) 495-552. | MR | Zbl
[2] , and , Complete analysis of phase transitions and ensemble equivalence for the Curie-Weiss-Potts model. J. Math. Phys. 46 (2005) 063301. | MR | Zbl
[3] and , Large deviations techniques and applications Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin 38 (2010). Corrected reprint of the second edition (1998). | MR | Zbl
[4] and , Large deviations for exchangeable random vectors. Ann. Probab. 20 (1992) 1147-1166. | MR | Zbl
[5] and , The Curie-Weiss model with dynamical external field. Markov Process. Related Fields 15 (2009) 1-30. | MR | Zbl
[6] and , A Weak Convergence Approach to the Theory of Large Deviations. Probab. Stat. John Wiley & Sons Inc., New York (1997). A Wiley-Interscience Publication. | MR | Zbl
[7] and , Moderate deviations for a class of mean-field models. Markov Process. Related Fields 10 (2004) 345-366. | MR | Zbl
[8] , Entropy, large deviations, and statistical mechanics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York 271 (1985). | MR | Zbl
[9] and , Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44 (1978) 117-139. | MR | Zbl
[10] , and , 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. | MR | Zbl
[11] , and , Metastates in mean-field models with random external fields generated by Markov chains. J. Stat. Phys. 146 (2012) 314-329. | MR | Zbl
[12] and , Dynamic random walks. Theory and applications. Elsevier B. V., Amsterdam (2006). | MR | Zbl
[13] and , Moderate Deviations for Random Field Curie-Weiss Models. J. Stat. Phys. 149 (2012) 701-721. | MR | Zbl
[14] , Ergodic Theory, vol. 2 of Adv. Math. Cambridge University Press, Cambridge (1983). | MR | Zbl
Cité par Sources :
