Approximate Ultrametricity for Random Measures and Applications to Spin Glasses

Abstract

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 two important conjectures regarding mixed p-spin glasses.

41 Pages, 1 figure

References in corpus (4)

• Complexity of random smooth functions on the high-dimensional sphere
• The Sherrington-Kirkpatrick model: an overview
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Cited by in corpus (9)

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• Thouless-Anderson-Palmer equations for the generic p-spin glass model
• Optimizing Mean Field Spin Glasses with External Field
• The Overlap Gap Property in Principal Submatrix Recovery
• Ultrametric Fitting by Gradient Descent
• On the overlap distribution of branching random walks
• Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function
• On the TAP equations via the cavity approach in the generic mixed $p$-spin models
• On Marginal Stability in Low Temperature Spherical Spin Glasses