Claussen G, Apolo L, Melchert O, Hartmann A K
Institut für Physik, Universität Oldenburg, Carl-von-Ossietzky Strasse, 26111 Oldenburg, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Nov;86(5 Pt 2):056708. doi: 10.1103/PhysRevE.86.056708. Epub 2012 Nov 19.
We consider the negative weight percolation (NWP) problem on hypercubic lattice graphs with fully periodic boundary conditions in all relevant dimensions from d=2 to the upper critical dimension d=6. The problem exhibits edge weights drawn from disorder distributions that allow for weights of either sign. We are interested in the full ensemble of loops with negative weight, i.e., nontrivial (system spanning) loops as well as topologically trivial ("small") loops. The NWP phenomenon refers to the disorder driven proliferation of system spanning loops of total negative weight. While previous studies where focused on the latter loops, we here put under scrutiny the ensemble of small loops. Our aim is to characterize-using this extensive and exhaustive numerical study-the loop length distribution of the small loops right at and below the critical point of the hypercubic setups by means of two independent critical exponents. These can further be related to the results of previous finite-size scaling analyses carried out for the system spanning loops. For the numerical simulations, we employed a mapping of the NWP model to a combinatorial optimization problem that can be solved exactly by using sophisticated matching algorithms. This allowed us to study here numerically exact very large systems with high statistics.
我们考虑在超立方晶格图上的负权重渗流（NWP）问题，该晶格图在从二维到上临界维度六维的所有相关维度上均具有完全周期性边界条件。该问题呈现出从无序分布中抽取的边权重，这些权重可以是正的也可以是负的。我们感兴趣的是具有负权重的环的完整集合，即非平凡（跨越系统）环以及拓扑平凡的（“小”）环。NWP现象指的是由无序驱动的总负权重跨越系统环的增殖。虽然先前的研究集中在后一种环上，但我们在此仔细研究小环的集合。我们的目标是通过两个独立的临界指数，利用这种广泛且详尽的数值研究来刻画超立方结构临界点及以下小环的环长分布。这些指数可以进一步与先前对跨越系统环进行的有限尺寸标度分析结果相关联。对于数值模拟，我们将NWP模型映射到一个组合优化问题，该问题可以通过使用复杂的匹配算法精确求解。这使我们能够在此对具有高统计量的非常大的系统进行精确的数值研究。
