Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Vector spin glasses with Mattis interaction II: non-convex high-temperature models

Hong-Bin Chen, Victor Issa

Abstract

This paper constitutes the second part of a two-paper series devoted to the systematic study of vector spin glass models whose energy function involves a spin glass part and a general Mattis interaction part. In this paper, we focus on models whose spin glass part does not satisfy the usual convexity assumption. In this case, the Parisi formula breaks down, and there are no known methods to fully identify the limit free energy. It was suggested in [arXiv:1906.08471] that the limit free energy may be described using the unique solution of a partial differential equation of Hamilton--Jacobi type. In the present paper, we prove the validity of this conjecture in the high-temperature regime and provide an explicit representation for the free energy in terms of critical points. Using the duality between the free energy and large deviation principles, one can then easily deduce from the previous result a large deviation principle for the mean magnetization as well as a representation for the free energy of spin glass models with additional Mattis interaction at high temperature. In the companion paper, we establish similar results at all temperatures for models whose spin glass part is assumed to satisfy the usual convexity assumption.

Vector spin glasses with Mattis interaction II: non-convex high-temperature models

Abstract

This paper constitutes the second part of a two-paper series devoted to the systematic study of vector spin glass models whose energy function involves a spin glass part and a general Mattis interaction part. In this paper, we focus on models whose spin glass part does not satisfy the usual convexity assumption. In this case, the Parisi formula breaks down, and there are no known methods to fully identify the limit free energy. It was suggested in [arXiv:1906.08471] that the limit free energy may be described using the unique solution of a partial differential equation of Hamilton--Jacobi type. In the present paper, we prove the validity of this conjecture in the high-temperature regime and provide an explicit representation for the free energy in terms of critical points. Using the duality between the free energy and large deviation principles, one can then easily deduce from the previous result a large deviation principle for the mean magnetization as well as a representation for the free energy of spin glass models with additional Mattis interaction at high temperature. In the companion paper, we establish similar results at all temperatures for models whose spin glass part is assumed to satisfy the usual convexity assumption.
Paper Structure (17 sections, 22 theorems, 198 equations)

This paper contains 17 sections, 22 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Preamble
  3. Setting
  4. Main results
  5. Recovering concrete models from the general setting
  6. Organization of the paper
  7. Acknowledgements
  8. Regularity of psi
  9. Short-time smooth solution of the Hamilton--Jacobi equation
  10. Proof of the main results
  11. Preliminaries
  12. Limit free energy and the large deviation principle
  13. Replica symmetric form
  14. Multi-species models
  15. Setting
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exists $\beta_\star \geqslant \frac{1}{2\sqrt{2}}$ such that for every $\beta < \beta_\star$ and almost every realization of $(W_{i,j})_{i,j}$, $(\chi^1_i)_i$ and $(\chi^2_i)_i$, the following holds. We have and the $\mathbb{R}^2$-valued mean magnetization $m_N$ satisfies a large deviation principle under $\langle \cdot \rangle_N$ with rate function $J$ defined for $m\in\mathbb{R}^2$ by wh

Theorems & Definitions (51)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3: Lower bound for $t_\star$
  • Theorem 1.4: Limit free energy
  • Remark 1.5: Concentration
  • Remark 1.6: Free energy of models without Mattis interaction
  • Definition 1.7: Large deviation principle
  • Theorem 1.8: Large deviations of $m_N$
  • Theorem 1.9: Replica symmetric form
  • Remark 1.10: Gaussian external field
  • ...and 41 more