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2009 One-dimensional random field Kac's model: weak large deviations principle
Pierre Picco, Enza Orlandi
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Electron. J. Probab. 14: 1372-1416 (2009). DOI: 10.1214/EJP.v14-662


We present a quenched weak large deviations principle for the Gibbs measures of a Random Field Kac Model (RFKM) in one dimension. The external random magnetic field is given by symmetrically distributed Bernouilli random variables. The results are valid for values of the temperature and magnitude of the field in the region where the free energy of the corresponding random Curie Weiss model has only two absolute minimizers. We give an explicit representation of the large deviation rate function and characterize its minimizers. We show that they are step functions taking two values, the two absolute minimizers of the free energy of the random Curie Weiss model. The points of discontinuity are described by a stationary renewal process related to the $h$-extrema of a bilateral Brownian motion studied by Neveu and Pitman, where $h$ depends on the temperature and magnitude of the random field. Our result is a complete characterization of the typical profiles of RFKM (the ground states) which was initiated in [2] and extended in [4].


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Pierre Picco. Enza Orlandi. "One-dimensional random field Kac's model: weak large deviations principle." Electron. J. Probab. 14 1372 - 1416, 2009.


Accepted: 16 June 2009; Published: 2009
First available in Project Euclid: 1 June 2016

zbMATH: 1191.60117
MathSciNet: MR2511287
Digital Object Identifier: 10.1214/EJP.v14-662

Primary: 60K35
Secondary: 82B20 , 82B43

Keywords: Kac potential , large deviations random walk , phase transition , random environment

Vol.14 • 2009
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