We study the complexity of approximating the partition function of dense Ising models in the critical regime. Recent work of Chen, Chen, Yin, and Zhang (FOCS 2025) established fast mixing at criticality, and even beyond criticality in a window of width $N^{-1/2}$. We complement these algorithmic results by proving nearly tight hardness bounds, thus yielding the first instance of a sharp scaling window for the computational complexity of approximate counting. Specifically, for the dense Ising model we show that approximating the partition function is computationally hard within a window of width $N^{-1/2+\varepsilon}$ for any constant $\varepsilon>0$. Standard hardness reductions for non-critical regimes break down at criticality due to bigger fluctuations in the underlying gadgets, leading to suboptimal bounds. We overcome this barrier via a global approach which aggregates fluctuations across all gadgets rather than requiring tight concentration guarantees for each individually. This new approach yields the optimal exponent for the critical window.

翻译：我们研究了临界状态下密集伊辛模型配分函数近似计算的复杂性。Chen、Chen、Yin与Zhang（FOCS 2025）的最新工作建立了临界点甚至超越临界点（在宽度为$N^{-1/2}$的窗口内）的快速混合性质。我们通过证明近乎紧致的下界来补充这些算法结果，从而首次揭示了近似计数计算复杂性的尖锐标度窗口。具体而言，对于密集伊辛模型，我们证明：对于任意常数$\varepsilon>0$，在宽度为$N^{-1/2+\varepsilon}$的窗口内，配分函数的近似计算是计算困难的。由于底层工具中更大的涨落，非临界态标准困难性规约在临界点失效，导致次优界。我们通过全局方法克服了这一障碍——该方法聚合所有工具的整体涨落，而非要求每个单独工具具有紧致集中性保证。这一新方法为临界窗口导出了最优指数。