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.
This work has been supported by Deutsche Forschungsgemeinschaft (DFG) - project number 432176920.
We acknowledge the Allianz für Hochleistungsrechnen Rheinland-Pfalz for granting us access to the High Performance Computing Elwetritsch, on which our numerical simulations have been performed. It is furthermore a pleasure to thank Yan V. Fyodorov, Giorgio Parisi, Timm Plefka, Federico Ricci-Tersenghi and Marius A. Schmidt for enlightening conversations.
"TAP equations are repulsive." Electron. Commun. Probab. 27 1 - 7, 2022. https://doi.org/10.1214/22-ECP505