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

2025-07-21

AI Summary

This work addresses the efficient estimation of the partition function for mean-field spin glasses (e.g., the Sherrington–Kirkpatrick model). Methodologically, it integrates mean-field statistical physics, zero-location techniques from complex analysis, and average-case complexity theory—marking the first such synthesis—to design a deterministic quasipolynomial-time algorithm. The algorithm achieves high-precision approximation of the partition function within the second-moment region and for almost all real inverse temperatures, covering the vast majority of the replica-symmetric phase in the complex plane; it applies uniformly to both Ising and spherical spin systems. Its key contribution lies in establishing a quantitative link between the geometric structure of complex zeros and algorithmic tractability, thereby overcoming the limitations of conventional randomized algorithms—namely, their dependence on random initialization and sensitivity to parameter tuning. This work introduces a new paradigm for deterministic approximation of partition functions in disordered systems.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Computational social science

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 nearly all inverse temperatures in the second moment regime. In particular, for the Sherrington--Kirkpatrick model, our algorithms succeed for almost 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.

Problem

• Design algorithms for estimating spin glass partition functions
• Study zeros of partition functions in disordered systems
• Apply methods to Sherrington-Kirkpatrick and Ising models

Innovation

• Deterministic quasipolynomial-time algorithms for partition function
• Study zeros of partition function for accuracy
• Applicable to spherical and Ising spin cases