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Disorder chaos in the spherical mean-field model. (English) Zbl 1357.60106

Summary: We study the problem of disorder chaos in the spherical mean-field model. It concerns the behavior of the overlap between two independently sampled spin configurations from two Gibbs measures with the same external parameters. The prediction states that if the disorders in the Hamiltonians are slightly decoupled, then the overlap will be concentrated near a constant value. Following Guerra’s replica symmetry breaking scheme, we establish this at the levels of the free energy and the Gibbs measure.

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] Chatterjee, S.: Disorder chaos and multiple valleys in spin glasses (2009). Preprint available at arXiv:0907.3381 · Zbl 1130.82019
[2] Chen, W.-K.: Disorder chaos in the Sherrington-Kirkpatrick model with external field. Ann. Probab. 41(5), 3345-3391 (2013) · Zbl 1303.60089 · doi:10.1214/12-AOP793
[3] Chen, W.-K.: The Aizenman-Sims-Starr scheme and Parisi formula for mixed \[p\] p-spin spherical models. Electron. J. Probab. 18(94), 1-14 (2013) · Zbl 1288.60127
[4] Chen, W.-K.: Chaos in the mixed even-spin models. Comm. Math. Phys. 328(3), 867-901 (2014) · Zbl 1294.82036 · doi:10.1007/s00220-014-1888-1
[5] Chen, W.-K.: Variational representations for the Parisi functional and the two-dimensional Guerra-Talagrand bound (2015). Preprint available at arXiv:1501.06635 · Zbl 1140.60355
[6] Chen, W.-K., Panchenko, D.: An approach to chaos in some mixed \[p\] p-spin models. Probab. Theory Rel. Fields 157(1-2), 389-404 (2013) · Zbl 1287.60115 · doi:10.1007/s00440-012-0460-1
[7] Panchenko, D.: A note on Talagrand’s positivity principle. Electron. Comm. Probab. 12, 401-410 (2007) · Zbl 1140.60355 · doi:10.1214/ECP.v12-1326
[8] Panchenko, D.: Chaos in temperature in generic \[2p2\] p-spin models (2015). Preprint available at arXiv:1502.03801 · Zbl 1357.82072
[9] Panchenko, D., Talagrand, M.: On the overlap in the multiple spherical SK models. Ann. Probab. 35(6), 2321-2355 (2007) · Zbl 1128.60086 · doi:10.1214/009117907000000015
[10] Rizzo, T.: Spin glasses: statics and dynamics: summer school, Paris 2007. Progr. Probab. 62, 143-157 (2009) · Zbl 1194.82101
[11] Talagrand, M.: Free energy of the spherical mean field model. Probab. Theory Rel. Fields 134(3), 339-382 (2006) · Zbl 1130.82019 · doi:10.1007/s00440-005-0433-8
[12] Talagrand, M.: Mean field models for spin glasses. Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge. A Series of Modern Surveys in Mathematics, vol. 55, Springer, Berlin (2011) · Zbl 1214.82002
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