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Disorder chaos in the Sherrington-Kirkpatrick model with external field. (English) Zbl 1303.60089
The author considers a spin system obtained by coupling two distinct Sherrington-Kirkpatrick (SK) models with the same temperature and external field whose Hamiltonians are correlated. The disorder chaos conjecture for the SK model states that the overlap under the corresponding Gibbs measure is essentially concentrated at a single value. In the absence of an external field, this statement was first confirmed by S. Chatterjee [“Disorder chaos and multiple valleys in spin glasses”, Preprint, arxiv:0907.3381]. In the present paper, using Guerra’s replica symmetry breaking bound [F. Guerra, Commun. Math. Phys. 233, No. 1, 1–12 (2003; Zbl 1013.82023)], the author proves that the SK model is also chaotic in the presence of the external field and the position of the overlap is determined by an equation related to Guerra’s bound and the Parisi measure. The present paper was partially motivated by the book by M. Talagrand [Mean field models for spin glasses. Volume II: Advanced replica-symmetry and low temperature. Berlin: Springer (2011; Zbl 1232.82005)].

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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References:
[1] Aizenman, M., Sims, R. and Starr, S. (2003). An extended variational principle for the SK spin-glass model. Phys. Rev. B 68 214403.
[2] Bolthausen, E. and Sznitman, A. S. (1998). On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247-276. · Zbl 0927.60071
[3] Brary, A. J. and Moore, M. A. (1987). Chaotic nature of the spin-glass phase. Phys. Rev. Lett. 58 57-60.
[4] Chatterjee, S. (2009). Disorder chaos and multiple valleys in spin glasses. Preprint. Available at . 0907.3381
[5] Derrida, B. (1981). Random-energy model: An exactly solvable model of disordered systems. Phys. Rev. B (3) 24 2613-2626. · Zbl 1323.60134
[6] Fisher, D. S. and Huse, D. A. (1986). Ordered phase of short range Ising spin glasses. Phys. Rev. Lett. 56 1601-1604.
[7] Guerra, F. (2003). Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 1-12. · Zbl 1013.82023
[8] Katzgraber, H. G. and Krza\ogonek kała, F. (2007). Temperature and disorder chaos in three-dimensional Ising spin glasses. Phys. Rev. Lett. 98 017201.
[9] McKay, S. R., Berker, A. N. and Kirkpatrick, S. (1982). Spin-glass behavior in frustrated Ising models with chaotic renormalization-group trajectories. Phys. Rev. Lett. 48 767-770.
[10] Panchenko, D. and Talagrand, M. (2007). On the overlap in the multiple spherical SK models. Ann. Probab. 35 2321-2355. · Zbl 1128.60086
[11] Ruelle, D. (1987). A mathematical reformulation of Derrida’s REM and GREM. Comm. Math. Phys. 108 225-239. · Zbl 0617.60100
[12] Sherrington, D. and Kirkpatrick, S. (1975). Solvable model of a spin glass. Phys. Rev. Lett. 35 1792-1796.
[13] Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221-263. · Zbl 1137.82010
[14] Talagrand, M. (2006). Parisi measures. J. Funct. Anal. 231 269-286. · Zbl 1117.82025
[15] Talagrand, M. (2007). Mean field models for spin glasses: Some obnoxious problems. In Spin Glasses. Lecture Notes in Math. 1900 63-80. Springer, Berlin. · Zbl 1116.82031
[16] Talagrand, M. (2011). Mean Field Models for Spin Glasses. Volume I : Basic Examples. Ergebnisse der Mathematik und Ihrer Grenzgebiete . 3. Folge. A Series of Modern Surveys in Mathematics [ Results in Mathematics and Related Areas . 3 rd Series. A Series of Modern Surveys in Mathematics ] 54 . Springer, Berlin. · Zbl 1214.82002
[17] Talagrand, M. (2011). Mean Field Models for Spin Glasses. Volume II : Advanced Replica-Symmetry and Low Temperature. Ergebnisse der Mathematik und Ihrer Grenzgebiete . 3. Folge. A Series of Modern Surveys in Mathematics [ Results in Mathematics and Related Areas . 3 rd Series. A Series of Modern Surveys in Mathematics ] 55 . Springer, Berlin. · Zbl 1232.82005
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