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On the local nature of the de Almeida-Thouless line for mixed \(p\)-spin glasses

Jean-Christophe Mourrat, Adrien Schertzer

TL;DR

This work shows that the generalized de Almeida-Thouless (AT) line proposed for mixed $p$-spin glasses does not, in general, characterize the replica symmetric (RS) phase. Leveraging a Hopf-Lax representation of the Parisi formula, the authors construct explicit counterexamples by augmenting an SK-like model with a high-degree $p$-spin term, demonstrating that AT can hold (in particular for all $\beta<1$ at zero field) even when the Parisi measure is not a Dirac mass. This reveals a fundamental limitation of local stability criteria for identifying RS in generic mixed models and clarifies that RS and AT cannot be equated in general. The SK model at zero field remains an open case for a complete RS–AT equivalence. These results refine our understanding of the RS region and motivate seeking global rather than purely local characterizations of replica symmetry in complex spin-glass landscapes.

Abstract

Jagannath and Tobasco~\cite{JagTob} proposed a generalized de Almeida-Thouless (AT) criterion aimed at characterizing the replica symmetric (RS) regime for a broad class of mixed \(p\)-spin glass models with Ising spins. In this paper, we show that this generalized AT condition does not characterize the RS regime in general. Using the Hopf-Lax representation of the Parisi formula, we construct explicit counterexamples within the class of mixed \(p\)-spin models. In particular, we exhibit a model in which the classical AT perturbation is performed around the unique minimizer of the RS free energy, and prove that even in this setting, the AT criterion fails to characterize the RS phase. By contrast, the validity of the classical AT condition for the Sherrington-Kirkpatrick model remains open.

On the local nature of the de Almeida-Thouless line for mixed \(p\)-spin glasses

TL;DR

This work shows that the generalized de Almeida-Thouless (AT) line proposed for mixed -spin glasses does not, in general, characterize the replica symmetric (RS) phase. Leveraging a Hopf-Lax representation of the Parisi formula, the authors construct explicit counterexamples by augmenting an SK-like model with a high-degree -spin term, demonstrating that AT can hold (in particular for all at zero field) even when the Parisi measure is not a Dirac mass. This reveals a fundamental limitation of local stability criteria for identifying RS in generic mixed models and clarifies that RS and AT cannot be equated in general. The SK model at zero field remains an open case for a complete RS–AT equivalence. These results refine our understanding of the RS region and motivate seeking global rather than purely local characterizations of replica symmetry in complex spin-glass landscapes.

Abstract

Jagannath and Tobasco~\cite{JagTob} proposed a generalized de Almeida-Thouless (AT) criterion aimed at characterizing the replica symmetric (RS) regime for a broad class of mixed -spin glass models with Ising spins. In this paper, we show that this generalized AT condition does not characterize the RS regime in general. Using the Hopf-Lax representation of the Parisi formula, we construct explicit counterexamples within the class of mixed -spin models. In particular, we exhibit a model in which the classical AT perturbation is performed around the unique minimizer of the RS free energy, and prove that even in this setting, the AT criterion fails to characterize the RS phase. By contrast, the validity of the classical AT condition for the Sherrington-Kirkpatrick model remains open.
Paper Structure (6 sections, 1 theorem, 39 equations)

This paper contains 6 sections, 1 theorem, 39 equations.

Table of Contents

  1. Introduction
  2. The model
  3. The Parisi formula
  4. The generalized AT line
  5. The AT criterion
  6. Proof of Theorem \ref{['theorem:main']}

Key Result

Theorem 1

There exist mixed $p$-spin models $\xi_0$ with $\xi_0"(0)>0$ such that

Theorems & Definitions (4)

  • Theorem 1
  • Remark 2
  • Remark 3
  • proof