半连续统几何中的渗流
Percolation in semicontinuum geometries
AI总结：
本文研究晶格与连续统方向混合的半连续统几何中重叠物体渗流，借助排除体积论证和Monte Carlo模拟分析超长方体、圆盘和平行四边形的阈值规律，并建立二维连续统与三角晶格的联系。
摘要：
我们研究重叠物体的渗流问题，其底层几何结构使得在D维空间中，一部分方向具有晶格结构，而其余方向具有连续统结构。由此产生的半连续统问题描述了平行层或通道中重叠形状的渗流，其中物体沿离散方向的放置存在位置约束。我们分析了若干半连续统渗流系统，例如超长方体，并特别关注二维和三维情形、圆盘以及平行四边形。通过将排除体积论证调整到半连续统情形，我们证明，对于超长方体的半连续统问题，在保持晶格结构的方向上固定超长方体边长时，渗流阈值始终与连续统方向上的边长无关。即使沿连续统方向的边长存在某种分布，该结果仍然成立。对于二维中的圆盘和平行四边形等其他半连续统模型，我们得到了当改变形状沿连续统方向的线性尺度时阈值变化的趋势。这些结果与相应的连续统模型和晶格模型的结果进行了比较。对于所考虑的二维和三维模型，我们利用Monte Carlo模拟验证了关于渗流阈值趋势和数值的排除体积预测。预测数值与模拟结果之间表现出非常好的一致性。半连续统情形还使我们能够建立二维连续统中重叠形状的渗流问题与三角晶格之间的联系。我们还验证了，在半连续统情形中，各向异性形状阈值的各向同性以及标准渗流普适类得以保持。
英文摘要：
We study percolation problems of overlapping objects where the underlying geometry is such that in D-dimensions, a subset of the directions has a lattice structure, while the remaining directions have a continuum structure. The resulting semicontinuum problem describes the percolation of overlapping shapes in parallel layers or lanes with positional constraints for the placement of the objects along the discrete directions. Several semicontinuum percolation systems are analyzed like hypercuboids with a particular focus on 2D and 3D cases, disks, and parallelograms. Adapting the excluded volume arguments to the semicontinuum setting, we show that for the semicontinuum problem of hypercuboids, for fixed side-lengths of the hypercuboids along the directions in which a lattice structure is maintained, the percolation threshold is always independent of the side-lengths along the continuum directions. The result holds even when there is a distribution for the side-lengths along the continuum directions. Trends in the variation of the thresholds, as we vary the linear measure of the shapes along the continuum directions, are obtained for other semicontinuum models like disks and parallelograms in 2D. The results are compared with those of corresponding continuum and lattice models. For the 2D and 3D models considered, using Monte Carlo simulations, we verify the excluded volume predictions for the trends and numerical values of the percolation thresholds. Very good agreement is seen between the predicted numerical values and the simulation results. The semicontinuum setting also allows us to establish a connection between the percolation problem of overlapping shapes in 2D continuum and the triangular lattice. We also verify that the isotropy of the threshold for anisotropic shapes and standard percolation universality class is maintained in the semicontinuum setting.
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