We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the ‘winding’ technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. Algorithms (SODA16), 514–527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice.

中文翻译：

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折线图上反铁磁伊辛模型的多项式时间逼近算法

我们提出了一种多项式时间马尔可夫链蒙特卡罗算法，用于估计任何折线图上反铁磁伊辛模型的配分函数。该算法的分析利用了由 McQuillan [CoRR abs/1301.2880 (2013)] 设计并由 Huang、Lu 和 Zhang [Proc. 第 27 次症状。光盘上。算法 (SODA16), 514–527]。我们表明，即使对于折线图，分配函数的精确计算也是#P-hard，这表明近似算法是可以预期的最佳算法。我们还展示了 Ising 模型的 Glauber 动力学在折线图上快速混合，例如 kagome 晶格。