标题： 连续体几何中的渗透现象 显示英文标题

标题： Percolation in semicontinuum geometries

摘要： 我们研究重叠物体的渗透问题，其中底层几何结构在D维空间中，部分方向具有格子结构，而其余方向具有连续结构。 这种半连续问题描述了在平行层或车道中具有位置约束的重叠形状的渗透。 分析了几种半连续渗透系统，如超长方体，特别关注2D和3D情况、圆盘和菱形。 将排除体积论证适应到半连续设置中，我们证明对于超长方体的半连续问题，在保持格子结构的方向上，超长方体的边长固定时，渗透阈值始终与连续方向上的边长无关。 即使在连续方向上存在边长分布时，该结果仍然成立。 对于其他半连续模型如2D中的圆盘和菱形，获得了当沿连续方向的形状线性度变化时阈值的变化趋势。 将结果与相应连续模型和格子模型的结果进行了比较。 对于所考虑的2D和3D模型，使用蒙特卡洛模拟，我们验证了关于渗透阈值趋势和数值的排除体积预测。 预测的数值与模拟结果之间显示出非常好的一致性。 半连续设置还使我们能够建立二维连续空间中重叠形状的渗透问题与三角形格子之间的联系。 我们还验证了各向异性形状的阈值各向同性和标准渗透普适类在半连续设置中得到保持。 显示英文摘要

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.