Approximate ultrametricity for random measures and applications to spin glasses. (English) Zbl 1361.60035
Summary: In this paper, we introduce a notion called “approximate ultrametricity,” which encapsulates the phenomenology of a sequence of random probability measures having supports that behave like ultrametric spaces insofar as they decompose into nested balls. We provide a sufficient condition for a sequence of random probability measures on the unit ball of an infinite-dimensional separable Hilbert space to admit such a decomposition, whose elements we call clusters. We also characterize the laws of the measures of the clusters by showing that they converge in law to the weights of a Ruelle probability cascade. These results apply to a large class of classical models in mean field spin glasses. We illustrate the notion of approximate ultrametricity by proving a conjecture of Talagrand regarding mixed \(p\)-spin glasses that is known to imply a prediction of Dotsenko-Franz-Mézard.
MSC:
| 60G57 | Random measures |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 60F05 | Central limit and other weak theorems |
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
| 82D30 | Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) |
Keywords:
random measures; approximate ultrametricity; Ruelle probability cascade; convergence in law; spin glassesReferences:
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