TAP equations are repulsive. (English) Zbl 1506.82016

Gufler, Stephan; Igelbrink, Jan Lukas; Kistler, Nicola

Electron. Commun. Probab. 27, Paper No. 63, 7 p. (2022).

Summary: We show that for low enough temperatures, but still above the AT line, the Jacobian of the TAP equations for the SK model has a macroscopic fraction of eigenvalues outside the unit interval. This provides a simple explanation for the numerical instability of the fixed points, which thus occurs already in high temperature. The insight leads to some algorithmic considerations on the low temperature regime.

MSC:

82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60B20	Random matrices (probabilistic aspects)
82D30	Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)

Keywords:

spin glasses; TAP equations

References:

[1]	Adhikari, Arka, Christian Brennecke, Per von Soosten, and Horng-Tzer Yau. Dynamical Approach to the TAP Equations for the Sherrington-Kirkpatrick Model. J. Stat. Phys. 183, 35 (2021). · Zbl 1473.82015
[2]	de Almeida J.R.L, and David J. Thouless. Stability of the Sherrington-Kirkpatrick solution of a spin glass model. J. Phys. A: Math. Gen. 11, pp. 983-990 (1978).
[3]	Anderson, Greg, Alice Guionnet and Ofer Zeitouni. An introduction to random matrices. Cambridge University Press (2011).
[4]	Aspelmeier, T., Blythe, R. A., Bray, A. J., and Moore, M. A.. Free-energy landscapes, dynamics, and the edge of chaos in mean-field models of spin glasses. Physical Review B, 74(18), 184411. (2006)
[5]	Bolthausen, Erwin. An iterative construction of solutions of the TAP equations for the Sherrington-Kirkpatrick model. Commun. Math. Phys. 325, 333-366 (2014). · Zbl 1288.82038
[6]	Bolthausen, Erwin. A Morita type proof of the replica-symmetric formula for SK. In: Statistical Mechanics of Classical and Disordered Systems. Springer (2019).
[7]	Celentano, Michael, Zhou Fan, and Song Mei. Local convexity of the TAP free energy and AMP convergence for Z2-synchronization. 2106.11428 (2021).
[8]	Chen, Wei-Kuo. On the Almeida-Thouless transition line in the SK model with centered Gaussian external field. Electron. Commun. Probab. 26: 1-9 (2021). · Zbl 1480.60287
[9]	Drazin, Michael P. and Emilie V. Haynsworth. Criteria for the reality of matrix eigenvalues. Math. Zeitschr. 78, 449-452 (1962). · Zbl 0117.01603
[10]	Sherrington, David, and Scott Kirkpatrick. Solvable model of a spin-glass. Physical review letters 35.26: 1792 (1975).
[11]	Talagrand, Michel. Mean field models for spin glasses. Springer (2011). · Zbl 1214.82002
[12]	Mézard, Marc, Giorgio Parisi, and Miguel A. Virasoro. Spin glass theory and beyond. World Scientific, Singapore (1987). · Zbl 0992.82500
[13]	Chen, Wei-Kuo, Dmitry Panchenko, and Eliran Subag. The generalized TAP free energy. 1812.05066 (2018).
[14]	Plefka, Timm. Convergence condition of the TAP equation for the infinite-ranged Ising spin glass model. Journal of Physics A: Math. Gen. 15.6 (1982): 1971-1978.
[15]	Tao, Terence. Topics in random matrix theory. AMS (2012). · Zbl 1256.15020
[16]	Tao, Terence, and Van Vu. Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge. Commun. Math. Phys. 298, 549-572 (2010). · Zbl 1202.15038
[17]	Thouless, David J., Philip W. Anderson, and Robert G. Palmer. Solution of ‘solvable model of a spin glass’. Philosophical Magazine 35.3 (1977): 593-601.