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Some rigorous results on the Sherrington-Kirkpatrick spin glass model. (English) Zbl 1108.82312

Commun. Math. Phys. 112, No. 1, 3-20 (1987); addendum 116, No. 3, 527 (1988).
Summary: We prove that in the high temperature regime \((T/J>1)\) the deviation of the total free energy of the Sherrington-Kirkpatrick (S-K) spin glass model from the easily computed \(\log Av(Z_N (\{\beta J\}))\) converges in distribution, as \(N\to \infty\), to a (shifted) Gaussian variable. Some weak results about the low temperature regime are also obtained.
In the addendum, the authors comment on one of the terms in the expression for the total free energy.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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References:

[1] Sherrington, D., Kirkpatrick, S.: Solvable model of a spin glass. Phys. Rev. Lett.35, 1792-1796 (1975) · doi:10.1103/PhysRevLett.35.1792
[2] Binder, K., Young, A.P.: Spin glasses: Experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys.54, 801-976 (1986) · doi:10.1103/RevModPhys.58.801
[3] Thouless, D.J., Anderson, P.W., Palmer, R.G.: Solution of solvable model of a spin glass. Philos. Mag.35, 593-601 (1977) · doi:10.1080/14786437708235992
[4] Feller, W.: An introduction to probability theory and its applications, Vol. II. New York: J. Wiley 1971 · Zbl 0219.60003
[5] Derrida, B.: Random energy model: An exactly solvable model of disordered systems. Phys. Rev. B24, 2613-2626 (1981) · Zbl 1323.60134
[6] Parisi, G.: The order parameter for spin glasses: A function on the interval 0-1. J. Phys. A13, 1101-1112 (1980)
[7] Palmer, R.G., Pond, C.M.: Internal field distribution in model spin glasses. J. Phys. F9, 1451-1459 (1979) · doi:10.1088/0305-4608/9/7/024
[8] Mezard, M., Parisi, G., Virasoro, M.: The replica solution without replicas. Europhys. Lett.1, 77-82 (1986) · doi:10.1209/0295-5075/1/2/006
[9] Ruelle, D.: A mathematical reformulation of Derrida’s REM and GREM. Commun. Math. Phys.108, 225-239 (1987) · Zbl 0617.60100 · doi:10.1007/BF01210613
[10] Mehta, M.L.: Random matrices. New York, London: Academic Press 1967 · Zbl 0925.60011
[11] Cohen, J., Kesten, H., Newman, C.: Random matrices and their applications. Contemp. Math., Vol. 50. Providence, RI: Am. Math. Soc. 1984
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