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

Wei-Kuo Chen, Si Tang

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.

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

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 -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.
Paper Structure (23 sections, 17 theorems, 201 equations)

This paper contains 23 sections, 17 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Related works
  3. Further results related to the TAP equations
  4. Organization of the paper
  5. Proof Sketch
  6. Notations
  7. Proof sketch of Theorem \ref{['thm:main']}
  8. Cavity method: A comparison
  9. The Cavity equation for $\langle \sigma_N\rangle_{N,\beta}^\alpha$
  10. Removal of the cavity spin constraint I
  11. Central limit theorem for the cavity field
  12. Proof of Theorem \ref{['thm:cavity']}
  13. Cavity computation for $X_{N,\beta}(\langle \sigma\rangle_{N,\beta}^\alpha)$
  14. Moment and truncation controls of $X_{N,\beta}(\langle \sigma \rangle_{N,\beta}^{\alpha})$ and $X_{N,\beta}(s^{\rho})$
  15. Removal of the cavity spin constraint II
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume that the model is generic. Then the conditional local magnetization $\langle \sigma \rangle^{\alpha}$ satisfies where $\epsilon \downarrow 0$ along a sequence such that $q_P-\epsilon$ is always a point of continuity for $\mu_P$, i.e., $\mu_P(\{q_P-\epsilon\})=0.$

Theorems & Definitions (34)

  • Theorem 1.1: TAP equations
  • Theorem 3.1: Cavity equation
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Remark 3.4
  • proof
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • ...and 24 more