Related papers: Free energy in the Sherrington-Kirkpatrick model w…

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There have been gaps found in the proofs. The paper is withdrawn until further notice.

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The Parisi formula for the free energy of the Sherrington-Kirkpatrick model is completed to a closed-form generating functional. We first find an integral representation for a solution of the Parisi differential equation and represent the…

This paper has been withdrawn by the authors due to the paper is far from complishment.

This paper has been withdrawn by the author due to errors.

We consider a spherical spin system with pure 2-spin spherical Sherrington-Kirkpatrick Hamiltonian with ferromagnetic Curie-Weiss interaction. The system shows a two-dimensional phase transition with respect to the temperature and the…

There have been comments on the starting paper, hep-th/0106074, which point out unclear motivation and definitions on noncommutative momentum introduced. Therefore, to give more clear presentation, this paper is withdrawn.

The paper has been withdrawn. The revised version is shown as cond-mat/0506752 and published in Condensed Matter Physics, vol.10, No.2(50)235-248(2007) (Lviv, Ukraine).

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This paper has been withdrawn due to an error in section 3.

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This paper has been withdrawn by the author. Much simpler proof of the main result was obtained which led to major changes in the presentation.

The paper has been withdrawn