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The Parisi ultrametricity conjecture

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arxiv 1112.1003 v2 pith:6CCE2JYK submitted 2011-12-05 math.PR

The Parisi ultrametricity conjecture

Dmitry Panchenko This is my paper
classification math.PR
keywords conjectureghirlanda-guerraidentitiesmodelsparisispinultrametricityball
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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. 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.

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    Difference spectra of two optimization-induced matrices from independent Gibbs samples form an explicit spectral transform of the Parisi overlap order parameter, while a single-matrix bulk is blind to replica symmetry...