On concavity of TAP free energy in the SK model

TL;DR

This paper investigates the spectral properties of the TAP free energy in the SK model, revealing its macroscopic concavity below the AT line and providing a spectral interpretation of the phase transition.

Contribution

It establishes the weak convergence of the Hessian's spectral distribution to a measure with negative support below the AT line, linking spectral properties to phase transitions.

Findings

Spectral distribution weakly converges to a measure with negative support below the AT line.

TAP free energy is macroscopically concave in the order parameter below the AT line.

Plefka's second condition is equivalent to all eigenvalues being negative in a simplified setting.

Abstract

We analyse 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, an that the support includes zero on the AT line. In this ``macroscopic'' sense, TAP free energy is concave in the order parameter of the theory, i.e. the random spin-magnetisations. This proves a spectral interpretation of the AT line. However, for specific magnetizations, the Hessian of the TAP free energy can have positive outlier eigenvalues. The question whether such outliers may also occur close to the TAP solutions is left open. In a simplified setting where the magnetizations are independent of the disorder, we prove that Plefka's second condition is equivalent to all eigenvalues being negative.