Color Coding for the Sherrington-Kirkpatrick Model

2026-10-07

AI Summary

This study addresses the computational bottleneck in approximating the partition function of the mean-field mixed p-spin model within the second-moment regime, where existing algorithms struggle to surpass quasi-polynomial time bounds. To overcome this limitation, we propose a deterministic polynomial-time algorithm that innovatively integrates color-coding techniques into the classical high-temperature expansion, coupled with rigorous analysis via zero-freeness theory. This approach achieves arbitrary-precision approximation of the partition function while reducing computational complexity to the polynomial level. Notably, its applicability fully encompasses the replica-symmetric region of the Sherrington–Kirkpatrick model and remains robust across typical disorder realizations. By significantly enhancing both computational efficiency and accuracy, this work provides a principled framework for tractable partition function estimation in disordered spin systems.

Abstract

We give a polynomial-time algorithm for approximating the partition function of mean-field mixed $p$-spin models to arbitrarily high accuracy throughout the second-moment regime. This improves the quasipolynomial-time algorithm of Bencs, Huang, Lee, Liu, and Regts (arXiv:2507.15616) and answers their open question. In particular, our result covers the entire replica-symmetric regime of the Sherrington--Kirkpatrick model. The algorithm is deterministic, runs in time polynomial in $n$ and $1/\varepsilon$, and succeeds for every typical realization of the disorder. Our main technical contribution is a new algorithmic application of color coding to the classical high-temperature expansion introduced by Aizenman, Lebowitz, and Ruelle in their study of fluctuations of the Sherrington--Kirkpatrick partition function. We combine this approach with the zero-freeness established by Bencs et al. (arXiv:2507.15616) to convert additive approximations into multiplicative ones.

Problem

• Sherrington-Kirkpatrick model
• partition function
• mean-field mixed p-spin models
• color coding
• replica-symmetric regime

Innovation

• Color Coding
• Sherrington-Kirkpatrick Model
• Partition Function Approximation
• High-Temperature Expansion
• Polynomial-Time Algorithm