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On differentiability of the Parisi formula
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Dmitry Panchenko, Texas A&M University
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Abstract
It was proved by Michel Talagrand in [10] that the Parisi
formula for the free energy
in the Sherrington-Kirkpatrick model is differentiable with
respect to inverse temperature parameter. We present a simpler proof of this
result by using approximate solutions in the Parisi formula and give one
example of application of the differentiability to prove non
self-averaging of the overlap outside of the replica symmetric region.
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Full text: PDF
Pages: 241-247
Published on: May 4, 2008
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Bibliography
- Aizenman, M.; Lebowitz, J. L.; Ruelle, D. Some rigorous results on the Sherrington-Kirkpatrick spin glass model. Comm. Math. Phys. 112 (1987), no. 1, 3--20. MR0904135 (88k:82104a)
- Guerra, Francesco. Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 (2003), no. 1, 1--12. MR1957729 (2003k:82048)
- Machta, J., Newman, C.M., Stein D.L. (2007) Percolation in the Sherrington-Kirk- patrick spin glass. Preprint, arXiv:0710.1399.
- Panchenko, Dmitry. A question about the Parisi functional. Electron. Comm. Probab. 10 (2005), 155--166 (electronic). MR2162815 (2006m:82063)
- Parisi, G. (1980) A sequence of approximate solutions to the S-K model for spin glasses. J. Phys. A 13, L-115.
- Pastur, L. A.; Shcherbina, M. V. Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model. J. Statist. Phys. 62 (1991), no. 1-2, 1--19. MR1105253 (92i:82063)
- Sherrington, D., Kirkpatrick, S. (1972) Solvable model of a spin glass. Phys. Rev. Lett. 35, 1792-1796.
- Talagrand, Michel. Spin glasses: a challenge for mathematicians.Cavity and mean field models.Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 46. Springer-Verlag, Berlin, 2003. x+586 pp. ISBN: 3-540-00356-8 MR1993891 (2005m:82074)
- Talagrand, Michel. On the meaning of Parisi's functional order parameter. C. R. Math. Acad. Sci. Paris 337 (2003), no. 9, 625--628. MR2017738 (2004k:82048)
- Talagrand, Michel. Parisi measures. J. Funct. Anal. 231 (2006), no. 2, 269--286. MR2195333 (2007h:82035)
- Talagrand, Michel. The Parisi formula. Ann. of Math. (2) 163 (2006), no. 1, 221--263. MR2195134 (2007m:82041)
- Toninelli, F.L. (2002) About the Almeida-Thouless transition line in the Sherrington-Kirkpatrick mean-field spin glass model, Europhys. Lett. 60, 764-767.
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Electronic Communications in Probability. ISSN: 1083-589X |
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