发表机构
MIT（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
我们提出一个确定性多项式时间算法，利用颜色编码和高温展开，在第二矩区域内高精度近似平均场混合$p$-自旋模型的配分函数，覆盖Sherrington-Kirkpatrick模型的复制对称区域，改进了先前拟多项式算法并回答了开放问题。
AI 中文摘要
我们给出一个多项式时间算法，用于在第二矩区域内以任意高精度近似平均场混合$p$-自旋模型的配分函数。这改进了Bencs、Huang、Lee、Liu和Regts（arXiv:2507.15616）的拟多项式时间算法，并回答了他们的开放问题。特别地，我们的结果覆盖了Sherrington-Kirkpatrick模型的整个复制对称区域。该算法是确定性的，运行时间在$n$和$1/\varepsilon$上为多项式，并且对每个典型的无序实现都成功。我们的主要技术贡献是将颜色编码应用于Aizenman、Lebowitz和Ruelle在研究Sherrington-Kirkpatrick配分函数涨落时引入的经典高温展开的一种新算法应用。我们将这种方法与Bencs等人（arXiv:2507.15616）建立的零自由性相结合，将加性近似转化为乘性近似。
英文摘要
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.
Comments33 pages
