On the REM approximation of TAP free energies

TL;DR

This paper develops a REM-like model for TAP free energies in the SK-model, revealing the origin of quadratic temperature corrections and formalizing cavity dynamics, with implications for hierarchical stability in evolving populations.

Contribution

It introduces an abstract REM-like framework for TAP free energies and clarifies the source of quadratic corrections in the SK-model's free energy.

Findings

Identifies the origin of quadratic temperature correction in TAP free energies.

Formalizes the true cavity dynamics acting on TAP solutions.

Provides a new perspective on hierarchical stability in population models.

Abstract

The free energy of TAP-solutions for the SK-model of mean field spin glasses can be expressed as a nonlinear functional of local terms: we exploit this feature in order to contrive abstract REM-like models which we then solve by a classical large deviations treatment. This allows to identify the origin of the physically unsettling quadratic (in the inverse of temperature) correction to the Parisi free energy for the SK-model, and formalizes the $\textit{true}$ cavity dynamics which acts on TAP-space, i.e. on the space of TAP-solutions. From a non-spin glass point of view, this work is the first in a series of refinements which addresses the stability of hierarchical structures in models of evolving populations.