AI 中文总结
该研究针对平面图上的自旋系统，证明硬核配分函数在小活动值下存在FPRAS，q≥4时q-着色计数近似为NP困难，还明确了2-自旋系统小外场下FPRAS的存在条件，证明思路来自GPT-5.6 Sol Ultra。
AI 中文摘要
我们证明，当活动值为足够小的常数时，硬核配分函数在平面图上存在完全多项式时间随机近似方案（FPRAS）。相反，我们证明对于任意常数$q\ge4$，近似计数平面图上的$q$-着色是NP困难的。我们还给出了平面图上2-自旋系统在足够小外场下存在FPRAS的完整特征。所有证明的核心思路均通过GPT-5.6 Sol Ultra得出。
英文摘要
We show that the hard-core partition function admits a fully polynomial-time randomised approximation scheme (FPRAS) on planar graphs when the activity is a sufficiently small constant. In contrast, we show that for any constant $q\ge 4$, approximately counting $q$-colourings in planar graphs is NP-hard. We also give a complete characterisation of when an FPRAS exists for a sufficiently small external field for 2-spin systems on planar graphs. The main ideas of all proofs were found using GPT-5.6 Sol Ultra.
Comments16 pages. v2: update a connection with [LO25], fix Cref issue
