On concentration properties of disordered Hamiltonians

Auffinger, Antonio; Chen, Wei-Kuo

doi:10.1090/proc/13864

On concentration properties of disordered Hamiltonians
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by Antonio Auffinger and Wei-Kuo Chen PDF

Proc. Amer. Math. Soc. 146 (2018), 1807-1815 Request permission

Abstract:

We present an elementary approach to concentration of disordered Hamiltonians. Assuming differentiability of the limiting free energy $F$ with respect to the inverse temperature $\beta$, we show that the Hamiltonian concentrates around the energy level $F’(\beta )$ under the free energy and Gibbs average.

References

Antonio Auffinger and Wei-Kuo Chen, Universality of chaos and ultrametricity in mixed $p$-spin models, Comm. Pure Appl. Math. 69 (2016), no. 11, 2107–2130. MR 3552010, DOI 10.1002/cpa.21617
Sourav Chatterjee, Absence of replica symmetry breaking in the random field Ising model, Comm. Math. Phys. 337 (2015), no. 1, 93–102. MR 3324156, DOI 10.1007/s00220-014-2269-5
Wei-Kuo Chen and Dmitry Panchenko, Some examples of quenched self-averaging in models with Gaussian disorder, Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017), no. 1, 243–258 (English, with English and French summaries). MR 3606741, DOI 10.1214/15-AIHP715
Stefano Ghirlanda and Francesco Guerra, General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity, J. Phys. A 31 (1998), no. 46, 9149–9155. MR 1662161, DOI 10.1088/0305-4470/31/46/006
Francesco Guerra and Fabio Lucio Toninelli, The thermodynamic limit in mean field spin glass models, Comm. Math. Phys. 230 (2002), no. 1, 71–79. MR 1930572, DOI 10.1007/s00220-002-0699-y
Marc Mézard, Giorgio Parisi, and Miguel Angel Virasoro, Spin glass theory and beyond, World Scientific Lecture Notes in Physics, vol. 9, World Scientific Publishing Co., Inc., Teaneck, NJ, 1987. MR 1026102
Dmitry Panchenko, The Ghirlanda-Guerra identities for mixed $p$-spin model, C. R. Math. Acad. Sci. Paris 348 (2010), no. 3-4, 189–192 (English, with English and French summaries). MR 2600075, DOI 10.1016/j.crma.2010.02.004
Dmitry Panchenko, The Sherrington-Kirkpatrick model, Springer Monographs in Mathematics, Springer, New York, 2013. MR 3052333, DOI 10.1007/978-1-4614-6289-7
G. Parisi, Infinite number of order parameters for spin-glasses. Phys. Rev. Lett., 43, 1754–1756 (1979)
A. Wayne Roberts and Dale E. Varberg, Convex functions, Pure and Applied Mathematics, Vol. 57, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0442824
Michel Talagrand, The Parisi formula, Ann. of Math. (2) 163 (2006), no. 1, 221–263. MR 2195134, DOI 10.4007/annals.2006.163.221
M. Talagrand, Mean field models for spin glasses. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 55, Springer-Verlag, Berlin (2011)