Abstract
We consider the problem of temperature dependence of the Gibbs states in two spin-glass models: Derrida's Random Energy Model and its analogue, where the random variables in the Hamiltonian are replaced by independent standard Brownian motions. For both of them we compute in the thermodynamic limit the overlap distribution ∑N i=1 σ i σ′ i /N∈[−1,1] of two spin configurations σ, σ′ under the product of two Gibbs measures, which are taken at temperatures T,T′ respectively. If T≠T′ are fixed, then at low temperature phase the results are different for these models: for the first one this distribution is D 0 δ 0+D 1 δ 1, with random weights D 0, D 1, while for the second one it is δ 0. We compute consequently the overlap distribution for the second model whenever T−T′→0 at different speeds as N→∞.
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REFERENCES
A. J. Bray and M. A. Moore, Chaotic nature of the spin-glass phase, Phys. Rev. Lett. 58:57-60 (1987).
I. Kondor, J. Phys. A 16:L127(1983).
I. Kondor and A. Véguö, Sensitivity of spin-glass order to temperature changes, J. Phys. A 34:L641, (2001).
T. Rizzo, Against chaos in temperature in mean-field spin-glass models, J. Phys. A 34:5531-5549 (2001).
A. Billoire and E. Marinari, Evidence against temperature chaos in mean-field and realistic spin glasses, J. Phys. A 33:L265(2000).
A. Billoire and E. Marinari (2002), Overlap among states at different temperatures in the SK model, cond-mat/0202473.
F. Krzakala and O. C. Martin, Chaotic temperature dependence in a model of spin glasses, cond-mat/0203449.
B. Derrida, Random energy model: Limit for a family of disordered models, Phys. Rev. Lett. 45:79-82 (1980).
B. Derrida, Random energy model: An exactly solvable model of disordered systems, Phys. Rev. B 24:2613-2626
T. Eisele, On a third order phase transition, Commun. Math. Phys. 90:125-159 (1983).
E. Olivieri and P. Picco, On the existence of thermodynamics for the random energy model, Commun. Math. Phys. 96:125-144 (1991).
A. Galvez, S. Martinez, and P. Picco, Fluctuations in Derrida's random energy model and generalised random energy models, J. Stat. Phys. 54:515-529 (1989).
T. C. Dorlas and J. R. Wedagedera, Large deviations and the random energy model, Int. J. Mod. Phys. B 15:1-15, 2001.
D. Ruelle, A mathematical reformulation of Derrida's REM and GREM, Commun Math. Phys. 108:225-239 (1987).
A. Bovier, I. Kurkova, and M. Löwe, Fluctuations of the free energy in the REM and the p‐spin SK-models, to appear in Ann. Probab. (2002)
A. Bovier, Statistical mechanics of disordered systems, MaPhySto Lecture Notes 10 (2001).
M. Talagrand, Mean-field models for spin glasses: A first course. Course given at Saint Flour Probability Summer School, 2000.
M. F. Kratz and P. Picco, On a representation of Gibbs measure for REM. Preprint. Samos 151 (2002).
Ch. M. Newman, Topics in Disordered Systems, Lectures in Mathematics, ETH Zürich (Bikhauser Verlag, Basel, 1997).
Ch. M. Newman and D. L. Stein, Thermodynamic chaos and the structure of short range spin glasses, in Mathematical Aspects of Spin Glasses and Neural Networks, Progress in Probability, A. Bovier and P. Picco, eds. (Birkhaüser, Boston, 1997).
Ch. M. Newman and D. L. Stein, Metastate approach to thermodynamic chaos. Phys. Rev. E 55:5194-5211 (1997).
M. R. Leadbetter, G. Lindgren, and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes (Springer, Berlin/Heidelberg/New York, 1983).
D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes (Springer-Verlag, 1988).
W. Feller, Introduction to the Probability Theory and Its Applications, Vol. I (Wiley, New York/London/Sidney, 1950).
E. Gardner and B. Derrida, The probability distribution of the partition function of the random energy model, J. Phys. A 22:1975-1981 (1989).
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Kurkova, I. Temperature Dependence of the Gibbs State in the Random Energy Model. Journal of Statistical Physics 111, 35–56 (2003). https://doi.org/10.1023/A:1022244721936
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DOI: https://doi.org/10.1023/A:1022244721936