负权渗流问题的上临界维数
The upper critical dimension of the negative-weight percolation problem
摘要：
通过数值模拟，我们调查的几何性质的循环超立方格图的尺寸d=2至7，其中边的权重是从一个分布，允许正负的权重。我们感兴趣的是总负权重的系统生成环的出现。由此产生的负权重渗流（NWP）问题是从根本上不同于传统的渗流，因为我们已经看到，在以前的研究中，这个模型的二维情况。在这里，我们描述了过渡的超立方系统，本研究的目的是得到一个把握的上临界尺寸du的NWP问题。对于数值模拟，我们采用的NWP模型的映射到一个组合优化问题，可以通过使用复杂的匹配算法来解决。我们通过类似于渗流理论中的观测量来表征循环，并进行有限尺寸标度分析，例如边长高达L=56个站点的3D超立方系统，以估计NWP现象的临界特性。我们发现我们的数值结果与NWP问题的上临界维数du =6是一致的。
英文摘要：
By means of numerical simulations we investigate the geometric properties of loops on hypercubic lattice graphs in dimensions d=2 through 7, where edge weights are drawn from a distribution that allows for positive and negative weights. We are interested in the appearance of system-spanning loops of total negative weight. The resulting negative-weight percolation (NWP) problem is fundamentally different from conventional percolation, as we have seen in previous studies of this model for the 2d case. Here, we characterize the transition for hypercubic systems, where the aim of the present study is to get a grip on the upper critical dimension d_u of the NWP problem. For the numerical simulations we employ a mapping of the NWP model to a combinatorial optimization problem that can be solved exactly by using sophisticated matching algorithms. We characterize the loops via observables similar to those in percolation theory and perform finite-size scaling analyses, e.g. 3d hypercubic systems with side length up to L=56 sites, in order to estimate the critical properties of the NWP phenomenon. We find our numerical results consistent with an upper critical dimension d_u=6 for the NWP problem.
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