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Fluctuations in random field Ising models

Seunghyun Lee, Nabarun Deb, Sumit Mukherjee

TL;DR

The paper studies fluctuations of linear statistics $T_n=\mathbf q^\top\boldsymbol\sigma$ under quadratic interaction Gibbs measures in the high-temperature regime, focusing on non-asymptotic Berry-Esseen bounds. It develops a unified LLN/CLT framework centered by the Mean-Field optimizer $\mathbf u$, with CLTs expressed in terms of an approximate eigenpair $\mathbf A_n\mathbf q\approx\lambda_n\mathbf q$ and variance $\upsilon_n/(1-\lambda_n\upsilon_n)$. The results cover RFIM on diverse graph ensembles (ER, regular, graphons) and Hopfield models, providing both quenched and annealed limits and extending CLTs to non-regular, possibly spin-glass–like settings. The methodology combines Stein's method of exchangeable pairs with Chevet-type concentration inequalities to handle general $\mathbf A_n$ and inhomogeneous fields, yielding explicit non-asymptotic error bounds and broad applicability to quadratic interaction models.

Abstract

This paper establishes a CLT for linear statistics of the form $\langle \mathbf{q},\boldsymbolσ \rangle$ with quantitative Berry-Esseen bounds, where $\boldsymbolσ$ is an observation from an exponential family with a quadratic form as its sufficient statistic, in the \enquote{high-temperature} regime. We apply our general result to random field Ising models with both discrete and continuous spins. To demonstrate the generality of our techniques, we apply our results to derive both quenched and annealed CLTs in various examples, which include Ising models on some graph ensembles of common interest (Erdős-Rényi, regular, dense bipartite), and the Hopfield spin glass model. Our proofs rely on a combination of Stein's method of exchangeable pairs and Chevet type concentration inequalities.

Fluctuations in random field Ising models

TL;DR

The paper studies fluctuations of linear statistics under quadratic interaction Gibbs measures in the high-temperature regime, focusing on non-asymptotic Berry-Esseen bounds. It develops a unified LLN/CLT framework centered by the Mean-Field optimizer , with CLTs expressed in terms of an approximate eigenpair and variance . The results cover RFIM on diverse graph ensembles (ER, regular, graphons) and Hopfield models, providing both quenched and annealed limits and extending CLTs to non-regular, possibly spin-glass–like settings. The methodology combines Stein's method of exchangeable pairs with Chevet-type concentration inequalities to handle general and inhomogeneous fields, yielding explicit non-asymptotic error bounds and broad applicability to quadratic interaction models.

Abstract

This paper establishes a CLT for linear statistics of the form with quantitative Berry-Esseen bounds, where is an observation from an exponential family with a quadratic form as its sufficient statistic, in the \enquote{high-temperature} regime. We apply our general result to random field Ising models with both discrete and continuous spins. To demonstrate the generality of our techniques, we apply our results to derive both quenched and annealed CLTs in various examples, which include Ising models on some graph ensembles of common interest (Erdős-Rényi, regular, dense bipartite), and the Hopfield spin glass model. Our proofs rely on a combination of Stein's method of exchangeable pairs and Chevet type concentration inequalities.

Paper Structure

This paper contains 14 sections, 18 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Notation
  4. Limit theory for quadratic interaction models
  5. Assumptions and Mean-Field approximation
  6. Weak laws and concentration inequalities
  7. Central Limit Theorems
  8. Proof overview and main challenges
  9. Random field Ising models
  10. Discussion
  11. Proof of main results
  12. Proof of results in \ref{['sec:isingrandom']}
  13. Proof of main lemmas
  14. Appendix: Proof of technical lemmas

Key Result

Lemma 2.1

Let $\mathbf{M}_n$ be a $n \times n$ symmetric matrix. Then, $\lVert \mathbf{M}_n\rVert \le \lVert \mathbf{M}_n\rVert_{r} \le \lVert \mathbf{M}_n\lVert_{\infty}$ for $r\in (2,\infty)$.

Theorems & Definitions (53)

  • Example 1.1: Random field Ising model
  • Definition 1.1: Exponential tilting
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.1: Eigenvalue-eigenvector
  • Remark 2.2: Delocalization
  • Remark 2.3: Necessity of \ref{['eq:smf']}
  • Theorem 2.5
  • ...and 43 more