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Bounds on the complexity of Replica Symmetry Breaking for spherical spin glasses


Authors: Aukosh Jagannath and Ian Tobasco
Journal: Proc. Amer. Math. Soc. 146 (2018), 3127-3142
MSC (2010): Primary 60K35, 82B44, 82D30, 49S05, 49N15; Secondary 49K15, 49N60
DOI: https://doi.org/10.1090/proc/13875
Published electronically: March 30, 2018
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Abstract: 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
Email: aukosh@math.harvard.edu

Ian Tobasco
Affiliation: Department of Mathematics, University of Michigan, 1854 East Hall, 530 Church Street, Ann Arbor, Michigan 48109
Email: itobasco@umich.edu

DOI: https://doi.org/10.1090/proc/13875
Received by editor(s): April 17, 2017
Published electronically: March 30, 2018
Communicated by: Zhen-Qing Chen
Article copyright: © Copyright 2018 American Mathematical Society

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