Moderate deviations for a Curie-Weiss model with dynamical external field
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 725-739

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.

DOI : 10.1051/ps/2012019
Classification : 60F10, 60K35, 82B44, 82B41, 60G50
Keywords: 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},
     year = {2013},
     publisher = {EDP Sciences},
     volume = {17},
     doi = {10.1051/ps/2012019},
     mrnumber = {3126159},
     zbl = {1290.60105},
     language = {en},
     url = {https://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  - https://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 https://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

[1] M. Costeniuc, R.S. Ellis and P. Tak-Hun Otto, Multiple critical behavior of probabilistic limit theorems in the neighborhood of a tricritical point. J. Stat. Phys. 127 (2007) 495-552. | Zbl | MR

[2] M. Costeniuc, R.S. Ellis and H. Touchette, Complete analysis of phase transitions and ensemble equivalence for the Curie-Weiss-Potts model. J. Math. Phys. 46 (2005) 063301. | Zbl | MR

[3] 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). | Zbl | MR

[4] I.H. Dinwoodie and S.L. Zabell, Large deviations for exchangeable random vectors. Ann. Probab. 20 (1992) 1147-1166. | Zbl | MR

[5] C. Dombry and N. Guillotin-Plantard, The Curie-Weiss model with dynamical external field. Markov Process. Related Fields 15 (2009) 1-30. | Zbl | MR

[6] 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. | Zbl | MR

[7] P. Eichelsbacher and M. Löwe, Moderate deviations for a class of mean-field models. Markov Process. Related Fields 10 (2004) 345-366. | Zbl | MR

[8] R.S. Ellis, Entropy, large deviations, and statistical mechanics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York 271 (1985). | Zbl | MR

[9] R.S. Ellis and C.M. Newman, Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44 (1978) 117-139. | Zbl | MR

[10] R.S. Ellis, C.M. Newman and J.S. Rosen, 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. | Zbl | MR

[11] M. Formentin, C. Külske and A. Reichenbachs, Metastates in mean-field models with random external fields generated by Markov chains. J. Stat. Phys. 146 (2012) 314-329. | Zbl | MR

[12] N. Guillotin-Plantard and R. Schott, Dynamic random walks. Theory and applications. Elsevier B. V., Amsterdam (2006). | Zbl | MR

[13] M. Löwe and R. Meiners, Moderate Deviations for Random Field Curie-Weiss Models. J. Stat. Phys. 149 (2012) 701-721. | Zbl | MR

[14] K. Petersen, Ergodic Theory, vol. 2 of Adv. Math. Cambridge University Press, Cambridge (1983). | Zbl | MR

Cité par Sources :