TAP equations are repulsive

TL;DR

This paper demonstrates that the Jacobian of TAP equations in the SK model exhibits eigenvalues outside the unit interval at certain temperatures, explaining numerical instability of fixed points even above the AT line.

Contribution

It reveals the spectral properties of the TAP Jacobian in the SK model, linking eigenvalues outside the unit interval to fixed point instability at specific temperatures.

Findings

Eigenvalues outside the unit interval occur above the AT line.

Numerical instability of fixed points is explained by spectral properties.

Insights into algorithmic behavior at low temperatures are discussed.

Abstract

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, also briefly discussed.