Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model |
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Authors: | L. A. Pastur M. V. Shcherbina |
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Affiliation: | (1) Mathematical Division of the Institute for Low Temperature Physics and Engineering of the Academy of Sciences of the Ukrainian SSR, 310164 47 Kharkov, USSR |
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Abstract: | We prove that ifNis the Sherrington-Kirkpatrick (SK) Hamiltonian and the quantity converges in the variance to a nonrandom limit asN, then the mean free energy of the model converges to the expression obtained by SK. Since this expression is known not to be correct in the low-temperature region, our result implies the non-self-averaging of the order parameter of the SK model. This fact is an important ingredient of the Parisi theory, which is widely believed to be exact. We also prove that the variance of the free energy of the SK model converges to zero asN, i.e., the free energy has the self-averaging property.See the Remarks after the proof of Theorem 1 on the validity of our results for more general distributions ofJij. |
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Keywords: | Spin glasses order parameter self-averaging |
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