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arXiv · 1412.7076

Approximate Ultrametricity for Random Measures and Applications to Spin Glasses

Aukosh Jagannath
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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.

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Aukosh Jagannath. 2014-12-22. Approximate Ultrametricity for Random Measures and Applications to Spin Glasses. https://doi.org/10.1002/cpa.21685

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