We study Derrida’s generalized random energy model (GREM) in the presence of uniform external field. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarse graining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarse-grained parts of the system emerging in the thermodynamic limit tend to have a certain optimal magnetization, as prescribed by the strength of the external field and by parameters of the GREM.
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December 2008
Research Article|
December 03 2008
Fluctuations of the partition function in the generalized random energy model with external field
Anton Bovier;
Anton Bovier
a)
1
Weierstraß-Institut für Angewandte Analysis und Stochastik
, Mohrenstraße 39, 10117 Berlin, Germany
2Institut für Mathematik,
Technische Universität Berlin
, Straße des 17 Juni 136, 10623 Berlin, Germany
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Anton Klimovsky
Anton Klimovsky
b)
2Institut für Mathematik,
Technische Universität Berlin
, Straße des 17 Juni 136, 10623 Berlin, Germany
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Anton Bovier
1,2,a)
Anton Klimovsky
2,b)
1
Weierstraß-Institut für Angewandte Analysis und Stochastik
, Mohrenstraße 39, 10117 Berlin, Germany
2Institut für Mathematik,
Technische Universität Berlin
, Straße des 17 Juni 136, 10623 Berlin, Germany
a)
Electronic mail: [email protected].
b)
Electronic mail: [email protected].
J. Math. Phys. 49, 125202 (2008)
Article history
Received:
May 12 2008
Accepted:
June 26 2008
Citation
Anton Bovier, Anton Klimovsky; Fluctuations of the partition function in the generalized random energy model with external field. J. Math. Phys. 1 December 2008; 49 (12): 125202. https://doi.org/10.1063/1.2962982
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