IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v164y2023icp160-182.html
   My bibliography  Save this article

On the concavity of the TAP free energy in the SK model

Author

Listed:
  • Gufler, Stephan
  • Schertzer, Adrien
  • Schmidt, Marius A.

Abstract

We analyze the Hessian of the Thouless–Anderson–Palmer (TAP) free energy for the Sherrington–Kirkpatrick model, below the de Almeida–Thouless line, evaluated in Bolthausen’s approximate solutions of the TAP equations. We show that the empirical spectral distribution weakly converges to a measure with negative support below the AT line, and that the support includes zero on the AT line. In this “macroscopic” sense, we show that TAP free energy is concave in the order parameter of the theory, i.e. the random spin-magnetizations. This proves a spectral interpretation of the AT line. We also find different magnetizations than Bolthausen’s approximate solutions at which the Hessian of the TAP free energy has positive outlier eigenvalues. In particular, when the magnetizations are assumed to be independent of the disorder, we prove that Plefka’s second condition is equivalent to all eigenvalues being negative. On this occasion, we extend the convergence result of Capitaine et al. (Electron. J. Probab. 16, no. 64, 2011) for the largest eigenvalue of perturbed complex Wigner matrices to the GOE.

Suggested Citation

  • Gufler, Stephan & Schertzer, Adrien & Schmidt, Marius A., 2023. "On the concavity of the TAP free energy in the SK model," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 160-182.
    • Handle: RePEc:eee:spapps:v:164:y:2023:i:c:p:160-182
    DOI: 10.1016/j.spa.2023.07.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414923001382
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2023.07.003?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:164:y:2023:i:c:p:160-182. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.