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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)
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