On the TAP equations via the cavity approach in the generic mixed $p$-spin models

TL;DR

This paper investigates the validity of TAP equations in the mixed p-spin models, demonstrating they are asymptotically satisfied by local magnetizations within pure states, thus linking to the model's energy landscape.

Contribution

It extends the understanding of TAP equations from the SK model to the more general mixed p-spin models using ultrametricity properties.

Findings

TAP equations are asymptotically valid in mixed p-spin models.

Conditional local magnetizations satisfy TAP equations in pure states.

Links TAP equations to the ultrametric structure of overlaps.

Abstract

In 1977, Thouless, Anderson, and Palmer (TAP) derived a system of consistent equations in terms of the effective magnetization in order to study the free energy in the Sherrington-Kirkpatrick (SK) spin glass model. The solutions to their equations were predicted to contain vital information about the landscapes in the SK Hamiltonian and the TAP free energy and moreover have direct connections to Parisi's replica ansatz. In this work, we aim to investigate the validity of the TAP equations in the generic mixed $p$-spin model. By utilizing the ultrametricity of the overlaps, we show that the TAP equations are asymptotically satisfied by the conditional local magnetizations on the asymptotic pure states.