Potts模型中传递矩阵的并行族树
Parallel family trees for transfer matrices in the Potts model
摘要：
Potts模型的传递矩阵方法的计算成本与\textit{into how many ways can two adjacent blocks of a lattice be connected}问题直接相关。解决这个问题会导致生成一组组合的晶格配置。这个集合定义了问题的\textit{configuration space}，它越小，传递矩阵法就能越快。在Potts模型中，条形格的一般转移矩阵方法的配置空间是Catalan数的数量级，导致$O(4^m)$的渐近代价，其中$m$是条形格的宽度。具有较小组态空间的转移矩阵方法确实存在，但它们对温度、自旋态的数量做出假设，或者限制晶格的拓扑结构以便工作。在本文中，我们提出了一个通用的和并行的传输矩阵方法，基于家庭树，使用一个子加泰罗尼亚配置空间的大小$O(3^m)$。改进是通过将原始的加泰罗尼亚配置集合分组到家族树的森林中来实现的，以这种方式，现在通过求解每个家族的根节点来计算问题的解决方案。因此，该算法的速度呈指数级增长，并且高度并行。另一个优点是，最终矩阵最终被压缩，不仅节省空间，而且使$(q,v)$上的数值计算比非压缩场景更快。实验结果表明，\textit{Parallel family trees (PFT)}策略在不同尺寸的带状网格上的运行速度确实比\textit{Catalan Parallel Method (CPM)}快，特别是在处理密集传输矩阵时。我们可以确认，并行实现的PFT算法是非常有效的和高效的大问题大小…
英文摘要：
The computational cost of transfer matrix methods for the Potts model is directly related to the problem of \textit{into how many ways can two adjacent blocks of a lattice be connected}. Answering this question leads to the generation of a combinatorial set of lattice configurations. This set defines the \textit{configuration space} of the problem, and the smaller it is, the faster the transfer matrix method can be. The configuration space of generic transfer matrix methods for strip lattices in the Potts model is in the order of the Catalan numbers, leading to an asymptotic cost of $O(4^m)$ with $m$ being the width of the strip. Transfer matrix methods with a smaller configuration space indeed exist but they make assumptions on the temperature, number of spin states, or restrict the topology of the lattice in order to work. In this paper we propose a general and parallel transfer matrix method, based on family trees, that uses a sub-Catalan configuration space of size $O(3^m)$. The improvement is achieved by grouping the original set of Catalan configurations into a forest of family trees, in such a way that the solution to the problem is now computed by just solving the root node of each family. As a result, the algorithm becomes exponentially faster and highly parallel. An additional advantage is that the final matrix ends up being compressed, not only saving space but also making numerical evaluation on $(q,v)$ faster than in a non-compressed scenario. Experimental results for different sizes of strip lattices show that the \textit{Parallel family trees (PFT)} strategy indeed runs exponentially faster than the \textit{Catalan Parallel Method (CPM)}, specially when dealing with dense transfer matrices. We can confirm that a parallel implementation of the PFT algorithm is highly effective and efficient for large problem sizes...
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