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In this paper 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.
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Aizenman, M., Contucci, P. (1998) On the stability of the quenched state in mean-field spin-glass models
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Ghirlanda, S., Guerra, F. (1998) General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity
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Guerra, F. (2003) Broken replica symmetry bounds in the mean field spin glass model
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Talagrand, M. (2003) Spin Glasses: a Challenge for Mathematicians. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics, Vol. 43. Springer-Verlag
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Talagrand, M. (2006) The Parisi formula
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Panchenko, D. (2010) The Ghirlanda-Guerra identities for mixed
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Talagrand, M. (2010) Construction of pure states in mean-field models for spin glasses
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Panchenko, D. (2011) Ghirlanda-Guerra identities and ultrametricity: An elementary proof in the discrete case
2011
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Panchenko, D. (2011) The Parisi formula for mixed
2011
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Talagrand, M. (2011) Mean-Field Models for Spin Glasses. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics, Vol. 54, 55. Springer-Verlag
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Panchenko, D. (2012) A unified stability property in spin glasses
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Arguin, L.-P., Aizenman, M. (2009) On the structure of quasi-stationary competing particles systems
2009
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Panchenko, D. (2010) A connection between Ghirlanda-Guerra identities and ultrametricity
2010
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