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

Stephan Gufler, Adrien Schertzer and Marius A. Schmidt

Stochastic Processes and their Applications, 2023, vol. 164, issue C, 160-182

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.

Keywords: Spin glasses; Sherrington and Kirkpatrick model; TAP free energy. (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414923001382
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:164:y:2023:i:c:p:160-182

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2023.07.003

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().