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Disorder Chaos in Short-Range, Diluted, and Lévy Spin Glasses

Wei-Kuo Chen, Heejune Kim, Arnab Sen

TL;DR

This work analyzes disorder chaos in spin glasses beyond the Edwards–Anderson setting by applying the Hermite spectral method to three related models: a mixed even $p$-spin short-range model on a bounded-degree hypergraph, a diluted mixed $p$-spin model, and a Lévy spin glass with index $oldsymbol{ m abla}oldsymbol{ m abla oldsymbol{ m olinebreak 2}}$ in $(1,2)$. The authors introduce an elementary algebraic equation for Fourier-Hermite coefficients of two-spin correlations, deriving geometric conditions that isolate contributing coefficients and enable chaos proofs under symmetric disorder for both continuous and discrete perturbations. They obtain explicit overlap bounds, showing that site-overlap chaos holds with decays that depend on graph growth rates and the Lévy index $oldsymbol{ m abla}$, thereby extending chaos results to non-Gaussian disorders and diluted graphs. By demonstrating the broad applicability of the Hermite spectral approach and connecting short-range, diluted, and heavy-tailed spin glasses, the work advances quantitative understanding of chaos phenomena and informs questions of algorithmic stability and sampling in complex disordered systems.

Abstract

In a recent breakthrough [arXiv:2301.04112], Chatterjee proved site disorder chaos in the Edwards-Anderson (EA) short-range spin glass model utilizing the Hermite spectral method. In this paper, we demonstrate the further usefulness of this Hermite spectral approach by extending the validity of site disorder chaos in three related spin glass models. The first, called the mixed even $p$-spin short-range model, is a generalization of the EA model where the underlying graph is a deterministic bounded degree hypergraph consisting of hyperedges with even number of vertices. The second model is the diluted mixed $p$-spin model, which is allowed to have hyperedges with both odd and even number of vertices. For both models, our results hold under general symmetric disorder distributions. The main novelty of our argument is played by an elementary algebraic equation for the Fourier-Hermite series coefficients for the two-spin correlation functions. It allows us to deduce necessary geometric conditions to determine the contributing coefficients in the overlap function, which in spirit is the same as the crucial Lemma 1 in [arXiv:2301.04112]. Finally, we also establish disorder chaos in the Lévy model with stable index $α\in (1, 2)$.

Disorder Chaos in Short-Range, Diluted, and Lévy Spin Glasses

TL;DR

This work analyzes disorder chaos in spin glasses beyond the Edwards–Anderson setting by applying the Hermite spectral method to three related models: a mixed even -spin short-range model on a bounded-degree hypergraph, a diluted mixed -spin model, and a Lévy spin glass with index in . The authors introduce an elementary algebraic equation for Fourier-Hermite coefficients of two-spin correlations, deriving geometric conditions that isolate contributing coefficients and enable chaos proofs under symmetric disorder for both continuous and discrete perturbations. They obtain explicit overlap bounds, showing that site-overlap chaos holds with decays that depend on graph growth rates and the Lévy index , thereby extending chaos results to non-Gaussian disorders and diluted graphs. By demonstrating the broad applicability of the Hermite spectral approach and connecting short-range, diluted, and heavy-tailed spin glasses, the work advances quantitative understanding of chaos phenomena and informs questions of algorithmic stability and sampling in complex disordered systems.

Abstract

In a recent breakthrough [arXiv:2301.04112], Chatterjee proved site disorder chaos in the Edwards-Anderson (EA) short-range spin glass model utilizing the Hermite spectral method. In this paper, we demonstrate the further usefulness of this Hermite spectral approach by extending the validity of site disorder chaos in three related spin glass models. The first, called the mixed even -spin short-range model, is a generalization of the EA model where the underlying graph is a deterministic bounded degree hypergraph consisting of hyperedges with even number of vertices. The second model is the diluted mixed -spin model, which is allowed to have hyperedges with both odd and even number of vertices. For both models, our results hold under general symmetric disorder distributions. The main novelty of our argument is played by an elementary algebraic equation for the Fourier-Hermite series coefficients for the two-spin correlation functions. It allows us to deduce necessary geometric conditions to determine the contributing coefficients in the overlap function, which in spirit is the same as the crucial Lemma 1 in [arXiv:2301.04112]. Finally, we also establish disorder chaos in the Lévy model with stable index .
Paper Structure (15 sections, 20 theorems, 125 equations, 3 figures)

This paper contains 15 sections, 20 theorems, 125 equations, 3 figures.

Table of Contents

  1. Introduction and Main Results
  2. Mixed Even $p$-spin short-range Model
  3. Diluted Mixed $p$-spin Model
  4. Lévy Model
  5. Related Results and Applications
  6. Open Problems
  7. Preliminaries
  8. Hypergraphs
  9. Hermite Polynomial Expansion
  10. Geometric Conditions for Vanishing Fourier-Hermite Coefficients
  11. Proof of Theorem \ref{['thm: mixed p spin disorder chaos']}
  12. Proof of Theorem \ref{['thm: diluted mixed p spin']}
  13. Exploration Process
  14. Proof of Theorem \ref{['chaosLevy']}
  15. Proof of item 2 in Remark \ref{['rem:Theorem_1.3']}

Key Result

Theorem 1.3

Consider the mixed even $p$-spin short-range model with the Hamiltonian hamiltonian. Let $t \ge 0$ and $\beta \in [0, \infty]$. Then, for both types of perturbations and for any $r \in \mathbb Z_+$, we have In particular, we can specialize into the following cases.

Figures (3)

  • Figure 1: Hypergraph $G$.
  • Figure 2: (Left panel): $v, v'$ belonging to the same hyperedge. (Right panel): $v, v'$ belonging to different hyperedges.
  • Figure 3: Left panel depicts Case 1 and right panel depicts Case 2.

Theorems & Definitions (46)

  • Definition 1.2: Two types of perturbations
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 36 more