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arXiv 2004.00920cond-mat.stat-mechcond-mat.dis-nn
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构造具有固定连通度的几何图的有效Monte-Carlo方法

An Efficient Monte-Carlo Method to Make a Geometric Graph with a Fixed Connectivity

Munetaka Sasaki

备注：
26 pages, 21 figures
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摘要：

本文提出了一种马尔可夫链蒙特卡罗（MCMC）方法来构造满足以下两个条件的几何图：（i）每个顶点的度固定为正整数$k$。(ii)位于$d$维超立方晶格上的两个顶点通过边连接的概率与$d_{ij}^{-α}$成比例，其中$d_{ij}$是两个顶点之间的距离，$α$是正指数。针对MCMC方法，我们引入了反向更新方法和基于列表的更新方法。由于MCMC方法和MCMC方法具有互补性，因此MCMC方法可以有效地更新图。我们还研究了铁磁伊辛模型定义的几何图形作为测试用例。结果表明，铁磁相变的性质与指数$α$密切相关.

英文摘要：

We present a Markov chain Monte-Carlo (MCMC) method to make a geometric graph which satisfies the following two conditions: (i) The degree of each vertex is fixed to a positive integer $k$. (ii) The probability that two vertices located on a $d$-dimensional hypercubic lattice are connected by an edge is proportional to $d_{ij}^{-α}$, where $d_{ij}$ is the distance between the two vertices and $α$ is a positive exponent. We introduce a reverse update method and a list-based update method for the MCMC method. The graph is updated efficiently by the MCMC method since the two update methods work complementarily. We also investigate a ferromagnetic Ising model defined on the geometric graph as a test case. As a result, we have confirmed that the nature of ferromagnetic transition significantly depends on the exponent $α$.

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