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The Parisi ultrametricity conjecture. (English) Zbl 1270.60060
Summary: We prove that the support of a random measure on the unit ball of a separable Hilbert space that satisfies the Ghirlanda-Guerra identities must be ultrametric with probability one. This implies the Parisi ultrametricity conjecture in mean-field spin glass models, such as the Sherrington-Kirkpatrick and mixed $$p$$-spin models, for which Gibbs measures are known to satisfy the Ghirlanda-Guerra identities in the thermodynamic limit.

##### MSC:
 60G57 Random measures 82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
##### Keywords:
invariance; spin glass models; ultrametricity
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##### References:
 [1] M. Aizenman and P. Contucci, ”On the stability of the quenched state in mean-field spin-glass models,” J. Statist. Phys., vol. 92, iss. 5-6, pp. 765-783, 1998. · Zbl 0963.82045 [2] L. Arguin and M. Aizenman, ”On the structure of quasi-stationary competing particle systems,” Ann. Probab., vol. 37, iss. 3, pp. 1080-1113, 2009. · Zbl 1177.60050 [3] L. N. Dovbysh and V. N. Sudakov, ”Gram-de Finetti matrices,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. $$($$LOMI$$)$$, vol. 119, pp. 77-86, 1982. · Zbl 0496.60018 [4] S. Ghirlanda and F. Guerra, ”General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity,” J. Phys. A, vol. 31, iss. 46, pp. 9149-9155, 1998. · Zbl 0953.82037 [5] F. Guerra, ”Broken replica symmetry bounds in the mean field spin glass model,” Comm. Math. Phys., vol. 233, iss. 1, pp. 1-12, 2003. · Zbl 1013.82023 [6] M. Mézard, G. Parisi, N. Sourlas, G. Toulouse, and M. Virasoro, ”On the nature of the spin-glass phase,” Phys. Rev. Lett., vol. 52, p. 1156, 1984. · Zbl 0968.82528 [7] M. Mézard, G. Parisi, N. Sourlas, G. Toulouse, and M. Virasoro, ”Replica symmetry breaking and the nature of the spin glass phase,” J. Physique, vol. 45, iss. 5, pp. 843-854, 1984. · Zbl 0968.82528 [8] M. Mézard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond, Teaneck, NJ: World Scientific Publishing Co., 1987. · Zbl 0992.82500 [9] D. Panchenko, ”A connection between the Ghirlanda-Guerra identities and ultrametricity,” Ann. Probab., vol. 38, iss. 1, pp. 327-347, 2010. · Zbl 1196.60167 [10] D. Panchenko, ”The Ghirlanda-Guerra identities for mixed $$p$$-spin model,” C. R. Math. Acad. Sci. Paris, vol. 348, iss. 3-4, pp. 189-192, 2010. · Zbl 1204.82036 [11] D. Panchenko, ”Ghirlanda-Guerra identities and ultrametricity: An elementary proof in the discrete case,” C. R. Math. Acad. Sci. Paris, vol. 349, iss. 13-14, pp. 813-816, 2011. · Zbl 1226.82030 [12] D. Panchenko, ”A unified stability property in spin glasses,” Comm. Math. Phys., vol. 313, iss. 3, pp. 781-790, 2012. · Zbl 1251.82051 [13] D. Panchenko, The Parisi formula for mixed $$p$$-spin models, 2011. · Zbl 1292.82020 [14] G. Parisi, ”Infinite number of order parameters for spin-glasses,” Phys. Rev. Lett., vol. 43, iss. 24, pp. 1754-1756, 1979. [15] G. Parisi, ”A sequence of approximate solutions to the S-K model for spin glasses,” J. Phys. A., vol. 13, p. l-115, 1980. [16] D. Sherrington and S. Kirkpatrick, ”Solvable model of a spin-glass,” Phys. Rev. Lett., vol. 35, pp. 1792-1796, 1975. [17] M. Talagrand, Spin Glasses: A Challenge for Mathematicians, New York: Springer-Verlag, 2003, vol. 46. · Zbl 1033.82002 [18] M. Talagrand, ”The Parisi formula,” Ann. of Math., vol. 163, iss. 1, pp. 221-263, 2006. · Zbl 1137.82010 [19] M. Talagrand, ”Construction of pure states in mean field models for spin glasses,” Probab. Theory Related Fields, vol. 148, iss. 3-4, pp. 601-643, 2010. · Zbl 1204.82037 [20] M. Talagrand, Mean Field Models for Spin Glasses. Volume I: Basic Examples, New York: Springer-Verlag, 2011, vol. 54. · Zbl 1214.82002 [21] M. Talagrand, Mean Field Models for Spin Glasses. Volume II: Advanced Replica-Symmetry and Low Temperature, , 2011, vol. 55. · Zbl 1232.82005
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