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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Bounds on the complexity of Replica Symmetry Breaking for spherical spin glasses
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by Aukosh Jagannath and Ian Tobasco PDF
Proc. Amer. Math. Soc. 146 (2018), 3127-3142 Request permission


In this paper, we study the Crisanti-Sommers variational problem, which is a variational formula for the free energy of spherical mixed $p$-spin glasses. We begin by computing the dual of this problem using a min-max argument. We find that the dual is a 1D problem of obstacle type, where the obstacle is related to the covariance structure of the underlying process. This approach yields an alternative way to understand Replica Symmetry Breaking at the level of the variational problem through topological properties of the coincidence set of the optimal dual variable. Using this duality, we give an algorithm to reduce this a priori infinite dimensional variational problem to a finite dimensional one, thereby confining all possible forms of Replica Symmetry Breaking in these models to a finite parameter family. These results complement the authors’ related results for the low temperature $\Gamma$-limit of this variational problem. We briefly discuss the analysis of the Replica Symmetric phase using this approach.
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Additional Information
  • Aukosh Jagannath
  • Affiliation: Department of Mathematics, Harvard University, Science Center, 1 Oxford Street, Cambridge, MA
  • MR Author ID: 989445
  • Email:
  • Ian Tobasco
  • Affiliation: Department of Mathematics, University of Michigan, 1854 East Hall, 530 Church Street, Ann Arbor, Michigan 48109
  • MR Author ID: 1057469
  • Email:
  • Received by editor(s): April 17, 2017
  • Published electronically: March 30, 2018
  • Communicated by: Zhen-Qing Chen
  • © Copyright 2018 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 146 (2018), 3127-3142
  • MSC (2010): Primary 60K35, 82B44, 82D30, 49S05, 49N15; Secondary 49K15, 49N60
  • DOI:
  • MathSciNet review: 3787372