AI 中文总结
该研究针对随机正则二分图上带均匀外场的反铁磁伊辛模型配分函数，推广硬核模型的近似方法，在特定参数条件下提出带截断的吉布斯采样算法，实现高效近似并证明其高概率性。
AI 中文摘要
我们针对随机正则二分图上带均匀外场的反铁磁伊辛模型的配分函数设计了随机近似方案。我们的算法将Kocurek、Oveis Gharan和Tjowasi（arXiv，2026）针对同一随机图模型上硬核模型在唯一性阈值之外的方法进行了推广。我们证明，只要λ被一个常数上界约束且λ(1−β)≲Δ⁻¹/²，高效随机算法就能以高概率近似配分函数。该算法首先截断二分图两侧规模过大的构型，随后从吉布斯分布中采样，采样条件为一侧或两侧的规模固定。为选择最优截断界，我们建立了随机正则二分图上吉布斯分布的集中性质，接着应用高维扩张并证明逐级传递定理，从而得到条件分布的快速采样器。
英文摘要
We design randomized approximation schemes for the partition function of antiferromagnetic Ising models with uniform external field on random regular bipartite graphs. Our algorithm generalizes the approach of Kocurek, Oveis Gharan and Tjowasi (arXiv, 2026) for hard-core models on the same random graph model beyond the uniqueness threshold. We show that, as long as $λ$ is upper bounded by a constant and $λ(1 - β) \lesssim Δ^{-1/2}$, an efficient randomized algorithm approximates the partition function with high probability. The algorithm first truncates configurations that are large on either side of the bipartition and then samples from Gibbs distributions conditioned on fixed sizes on one or both sides. To choose an optimal truncation bound, we establish concentration properties of the Gibbs distribution on random regular bipartite graphs. Then we apply high-dimensional expansion and prove trickle-down theorems to obtain fast samplers for the conditioned distributions.
