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On the GREM approximation of TAP free energies

Giulia Sebastiani, Marius Alexander Schmidt

Abstract

We establish both a Boltzmann-Gibbs principle and a Parisi formula for the limiting free energy of an abstract GREM (Generalized Random Energy Model) which provides an approximation of the TAP (Thouless-Anderson-Palmer) free energies associated to the Sherrington-Kirkpatrick (SK) model.

On the GREM approximation of TAP free energies

Abstract

We establish both a Boltzmann-Gibbs principle and a Parisi formula for the limiting free energy of an abstract GREM (Generalized Random Energy Model) which provides an approximation of the TAP (Thouless-Anderson-Palmer) free energies associated to the Sherrington-Kirkpatrick (SK) model.
Paper Structure (4 sections, 3 theorems, 83 equations)

This paper contains 4 sections, 3 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Definitions and results
  3. The Boltzmann-Gibbs principle: proof of Theorem \ref{['gibbs']}
  4. The Parisi principle: proof of Theorem \ref{['parisi_grem_tap_teor']}

Key Result

Theorem 2.1

Let ${\bf \Phi}: \mathcal{M}^+_1(\mathbb{R}^n)\to \mathbb{R}$ be a continous functional and consider Then the limit $\lim_{N\to\infty} f_N$ exists, is non-random and given by with

Theorems & Definitions (6)

  • Remark 1.1
  • Theorem 2.1: Boltzmann-Gibbs principle
  • Theorem 2.2: Parisi principle
  • Remark 2.3
  • Lemma 4.1
  • proof