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Existence of Two-Step Replica Symmetry Breaking for the Spherical Mixed p-Spin Glass at Zero Temperature

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Abstract

We provide the first examples of two-step replica symmetry breaking (2-RSB) models for the spherical mixed p-spin glass at zero temperature. Precisely, we show that for a certain class of mixtures, the Parisi measure at zero temperature is purely atomic and has exactly three distinct points in its support. We then derive a few consequences for the topology of the random landscape in these cases. Our main result also provides a negative answer to a question raised in 2011 by Auffinger and Ben Arous about the classification of pure-like and full mixture models.

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Acknowledgment

We would like to thank Wei-Kuo Chen for helpful discussions in the initial stage of this project. We also thank the referees for many suggestions, which have helped to improve the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Northwestern University, Evanston, USA

    Antonio Auffinger & Qiang Zeng

Authors
  1. Antonio Auffinger
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  2. Qiang Zeng
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Correspondence to Antonio Auffinger.

Additional information

Communicated by H. Spohn

Antonio Auffinger: Research partially supported by NSF Grant CAREER DMS-1653552 and NSF Grant DMS-1517894.

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Auffinger, A., Zeng, Q. Existence of Two-Step Replica Symmetry Breaking for the Spherical Mixed p-Spin Glass at Zero Temperature. Commun. Math. Phys. 370, 377–402 (2019). https://doi.org/10.1007/s00220-018-3252-3

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  • DOI: https://doi.org/10.1007/s00220-018-3252-3

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