The Parisi ultrametricity conjecture
By Dmitry Panchenko. 2011
234 citations recorded by the source corpus.
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.
Subfields: Condensed Matter Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/42a7b76d5f5dccfbeaa7d93d9b06fd2c14017bab
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In Condensed Matter Physics
In Statistics and Probability
Citations in the corpus
- A hierarchical version of the de Finetti and Aldous-Hoover representations (2013, background): “The main result in [14] proves precisely the hierarchical exchangeability (11) for the ra ndom variablesXα,i that represent the magnetization of theith spin inside the pure state α, and the tree structure as above stems from the ultrametric organization of the pure states in the Paris i n atz, which”
- The Parisi formula for mixed $p$-spin models (2011, methodology): “Using the main result in [18], which yields that under a small perturbation of the Hamiltonian the support of the Gibbs measure in these models is indeed asymptotically ultrametric, we prove the Parisi formula for general mixed p-spin models that include odd p-spin interactions as well.”