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On zeros and algorithms for disordered systems: mean-field spin glasses

Ferenc Bencs, Brice Huang, Daniel Z. Lee, Kuikui Liu, Guus Regts

TL;DR

This work establishes a principled connection between zero-free regions of the mean-field spin-glass partition function and efficient counting of these disordered systems. By introducing an analytically tractable reweighting factor and applying Jensen’s formula, the authors prove zero-free disks for ${Z_{m{G}}(eta)}$ in the second-moment regime and design deterministic Barvinok-type interpolation algorithms that approximate ${Z_{m{G}}(eta)}$ to any prescribed accuracy within a quasipolynomial running time. The approach is rigorously developed for both spherical and Ising mean-field spin models and yields a rigorous validation of a Plefka/ALR-style expansion up to ${O}( ext{log} n)$ corrections in the second-moment regime, with precise Curie–Weiss bounds feeding the analysis. The results illuminate the landscape of computational tractability in disordered systems, bridging phase-transition thresholds and algorithmic feasibility, and point to future work on sampling, external fields, and zero-free regions beyond the current RS window.

Abstract

Spin glasses are fundamental probability distributions at the core of statistical physics, the theory of average-case computational complexity, and modern high-dimensional statistical inference. In the mean-field setting, we design deterministic quasipolynomial-time algorithms for estimating the partition function to arbitrarily high accuracy for all inverse temperatures in the second moment regime. In particular, for the Sherrington--Kirkpatrick model, our algorithms succeed for the entire replica-symmetric phase. To achieve this, we study the locations of the zeros of the partition function. Notably, our methods are conceptually simple, and apply equally well to the spherical case and the case of Ising spins.

On zeros and algorithms for disordered systems: mean-field spin glasses

TL;DR

This work establishes a principled connection between zero-free regions of the mean-field spin-glass partition function and efficient counting of these disordered systems. By introducing an analytically tractable reweighting factor and applying Jensen’s formula, the authors prove zero-free disks for in the second-moment regime and design deterministic Barvinok-type interpolation algorithms that approximate to any prescribed accuracy within a quasipolynomial running time. The approach is rigorously developed for both spherical and Ising mean-field spin models and yields a rigorous validation of a Plefka/ALR-style expansion up to corrections in the second-moment regime, with precise Curie–Weiss bounds feeding the analysis. The results illuminate the landscape of computational tractability in disordered systems, bridging phase-transition thresholds and algorithmic feasibility, and point to future work on sampling, external fields, and zero-free regions beyond the current RS window.

Abstract

Spin glasses are fundamental probability distributions at the core of statistical physics, the theory of average-case computational complexity, and modern high-dimensional statistical inference. In the mean-field setting, we design deterministic quasipolynomial-time algorithms for estimating the partition function to arbitrarily high accuracy for all inverse temperatures in the second moment regime. In particular, for the Sherrington--Kirkpatrick model, our algorithms succeed for the entire replica-symmetric phase. To achieve this, we study the locations of the zeros of the partition function. Notably, our methods are conceptually simple, and apply equally well to the spherical case and the case of Ising spins.

Paper Structure

This paper contains 36 sections, 36 theorems, 277 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Comparison with Other Thresholds
  4. Prior Work
  5. Free Energy Formulas
  6. The Plefka and Aizenman--Lebowitz--Ruelle Expansions
  7. Localization and Markov Chains
  8. Approximate Counting from Zero-Freeness
  9. Acknowledgements
  10. Techniques and Overview of the Paper
  11. Counting Zeros via Jensen's Formula
  12. Reweighting the Partition Function
  13. Reduction from Zero-Counting to Second Moment
  14. Preliminaries
  15. Evaluation of Reweighted Second Moment
  16. ...and 21 more sections

Key Result

Theorem 1.1

For any $0 < \varepsilon,\eta < 1$, there exists $C = \Theta(1/\varepsilon)$ and $\delta = \delta(\varepsilon,n) > 0$ tending to $0$ as $n \to \infty$ such that where $\widehat{P}(\beta)$ denotes the Taylor polynomial of $\beta \mapsto \log Z_{\bm{G}}(\beta)$ of degree $C\log(n/\eta)$. In particular, there is a deterministic algorithm running in time $n^{C \log(n/\eta)}$ which, on input $\bm{G},\

Theorems & Definitions (75)

  • Definition 1: Second Moment Regime
  • Theorem 1.1: Algorithms
  • Theorem 1.2: Zeros
  • Theorem 2.1: Jensen's Formula
  • Lemma 2.2
  • Lemma 2.3
  • Remark 1
  • proof : Proof of \ref{['lem:jensen-formula-second-moment']}
  • proof : Proof of \ref{['lem:no-reweight-reduction-to-curie-weiss']}
  • Proposition 2.4
  • ...and 65 more