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Vector spin glasses with Mattis interaction I: the convex case

Hong-Bin Chen, Victor Issa

Abstract

This paper constitutes the first 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 satisfies the usual convexity assumption. We identify the limit free energy via a Parisi-type formula and prove a large deviation principle for the mean magnetization. The proof is remarkably simple and short compared to previous approaches; it relies on treating the Mattis interaction as a parameter of the model. In the companion paper, we establish similar results in the high-temperature regime for models whose spin glass part is not assumed to satisfy the usual convexity assumption.

Vector spin glasses with Mattis interaction I: the convex case

Abstract

This paper constitutes the first 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 satisfies the usual convexity assumption. We identify the limit free energy via a Parisi-type formula and prove a large deviation principle for the mean magnetization. The proof is remarkably simple and short compared to previous approaches; it relies on treating the Mattis interaction as a parameter of the model. In the companion paper, we establish similar results in the high-temperature regime for models whose spin glass part is not assumed to satisfy the usual convexity assumption.
Paper Structure (9 sections, 7 theorems, 55 equations)

This paper contains 9 sections, 7 theorems, 55 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. Identification of the limit free energy with linear Mattis interaction
  9. Proofs of main results

Key Result

Theorem 1.1

Almost surely over the randomness of $(W_{i,j})_{i,j}$ and $(\chi_i)_i$, we have where the infimum is taken over $m \in \mathbb{R}$ and the supremum over $x \in \mathbb{R}$ and $p : [0,1) \to \mathbb{R}_+$ bounded increasing càdlàg paths. Furthermore, still almost surely over the randomness of $(W_{i,j})_{i,j}$ and $(\chi_i)_i$, the random variable $m_N$ satisfies a large deviat Here $\phi^*(m) =

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3: Large deviation principle
  • Theorem 1.4: LDP for the magnetization
  • Proposition 2.1
  • proof : Sketch of proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 3.1
  • ...and 5 more