基于Kac--Ward理论的平面伊辛自旋玻璃的精确自回归采样
Exact autoregressive sampling of planar Ising spin glasses via the Kac--Ward theory
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中文总结 AI 辅助
本文基于Kac--Ward理论提出平面伊辛自旋玻璃的精确自回归采样算法，计算成本为O(N^(5/2))，为神经自回归采样器提供精确基准。
中文摘要 AI 辅助
从自旋玻璃的玻尔兹曼分布中进行精确采样仍是一项未解决的挑战：马尔可夫链蒙特卡罗方法存在临界减慢和亚稳态捕获问题，而变分自回归网络等现代神经自回归采样器是近似的，在缺乏精确参考样本的情况下无法进行严格基准测试。本文提出一种基于Kac--Ward理论的平面伊辛自旋玻璃精确自回归采样算法。在链式法则分解下，依次固定自旋会产生边界局域外场，破坏了精确评估所需的零场结构。通过用保平面性的辅助自旋构造对这些场进行编码，将条件配分函数映射到扩展的零场伊辛模型，并利用Kac--Ward行列式公式进行精确评估。该方法生成严格独立同分布的样本，具有精确的归一化似然，对于N个自旋的计算成本为O(N^(5/2))，从而为神经自回归采样器提供了精确的基准测试基线。
英文摘要
Exact sampling from the Boltzmann distribution of spin glasses remains an outstanding challenge: Markov chain Monte Carlo methods suffer from critical slowing down and metastable trapping, while modern neural autoregressive samplers such as variational autoregressive networks are approximate and, in the absence of exact reference samples, cannot be rigorously benchmarked. Here we present an exact autoregressive sampling algorithm for planar Ising spin glasses based on the Kac--Ward theory. Under the chain-rule factorization, sequentially fixing spins induces boundary-localized external fields, which destroy the zero-field structure required for exact evaluation. By encoding these fields with a planarity-preserving auxiliary spin construction, the conditional partition functions are mapped to an extended zero-field Ising model and exactly evaluated using the Kac--Ward determinant formula. The method generates strictly independent and identically distributed samples with exact normalized likelihoods at a computational cost of $\mathcal{O}(N^{5/2})$ for $N$ spins, thereby providing an exact baseline for benchmarking neural autoregressive samplers.
