三角格的抽样染色
Sampling Colourings of the Triangular Lattice
摘要：
我们表明，适当的9着色的三角形晶格上的Glauber动力学是迅速混合，这使得有效的采样。因此，有一个完全多项式随机近似计划（FPRAS）计数适当的9-着色的三角晶格。适当的着色对应于零温反铁磁Potts模型中的配置。我们证明了由三角晶格的适当9-着色组成的自旋系统具有强的空间混合。这意味着存在唯一的无限体积吉布斯分布，这是统计物理学中研究的一个重要性质。我们的结果建立在Goldberg，Martin和Paterson以前的工作基础上，他们在三角晶格上对10种颜色显示了类似的结果。他们的工作之前，萨拉斯和索卡尔的11色的结果。这两种证明都依赖于计算辅助，我们的9色证明也是如此。我们使用了随机启发法来指导我们获得严格的结果。
英文摘要：
We show that the Glauber dynamics on proper 9-colourings of the triangular lattice is rapidly mixing, which allows for efficient sampling. Consequently, there is a fully polynomial randomised approximation scheme (FPRAS) for counting proper 9-colourings of the triangular lattice. Proper colourings correspond to configurations in the zero-temperature anti-ferromagnetic Potts model. We show that the spin system consisting of proper 9-colourings of the triangular lattice has strong spatial mixing. This implies that there is a unique infinite-volume Gibbs distribution, which is an important property studied in statistical physics. Our results build on previous work by Goldberg, Martin and Paterson, who showed similar results for 10 colours on the triangular lattice. Their work was preceded by Salas and Sokal's 11-colour result. Both proofs rely on computational assistance, and so does our 9-colour proof. We have used a randomised heuristic to guide us towards rigourous results.
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