论文标题
自我避免的步行和多边形穿越广场和六角形晶格
Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices
论文作者
论文摘要
我们已经分析了最近扩展的系列，该系列的避免自我步行的数量（锯）$ C_L（1）$，该$ c_l（1）$跨越了$ l \ times l $ square之间。众所周知，此类步行的数量已成长为$λ_s^{l^2}。$我们已经根据几位作者提供的其他系列系数和精制的分析技术，使$λ_s的估计更加精确。 We estimate that $λ_S = 1.7445498 \pm 0.0000012.$ We have also studied the subdominant behaviour, and conjecture that $$ C_L(1) \sim λ_S^{L^2+bL+c}\cdot L^g,$$ where $b=-0.04354 \pm 0.0001,$ $c=0.5624 \pm 0.0005,$ $ g = 0.000 \ pm 0.005。$ 我们实现了一种非常有效的算法，用于在正方形和六角形晶格上枚举路径，以利用最小的完美哈希功能以及对路径数量计数的数组的原位内存更新。 使用该算法，我们扩展了跨正方形晶格的方形晶格和自避免的多边形（SAPS）的锯系列。已知这些也将成长为$λ_s^{l^2}。$ sub -dominant项$λ^b $与越过广场的锯相同，而指数$ g = 1.75 \ pm 0.01 $ 0.01 $ for Spanning Saws和$ g = -0.500 \ pm pm 0.005 $ 0.005 $ for SAPS saps saps。 我们还研究了六角形晶格上的类似问题，并为许多几何形状生成了序列。特别是，我们研究了六角形晶格上穿过物提，三角形和方形结构域的锯子和saps，以及跨越菱形的锯。我们估计，类似的生长常数$λ_H= 1.38724951 \ pm 0.00000005，$，因此比方格的更精确的估计值。我们还给出了次级主导术语的估计。
We have analysed the recently extended series for the number of self-avoiding walks (SAWs) $C_L(1)$ that cross an $L \times L$ square between diagonally opposed corners. The number of such walks is known to grow as $λ_S^{L^2}.$ We have made more precise the estimate of $λ_S,$ based on additional series coefficients provided by several authors, and refined analysis techniques. We estimate that $λ_S = 1.7445498 \pm 0.0000012.$ We have also studied the subdominant behaviour, and conjecture that $$ C_L(1) \sim λ_S^{L^2+bL+c}\cdot L^g,$$ where $b=-0.04354 \pm 0.0001,$ $c=0.5624 \pm 0.0005,$ and $g=0.000 \pm 0.005.$ We implemented a very efficient algorithm for enumerating paths on the square and hexagonal lattices making use of a minimal perfect hash function and in-place memory updating of the arrays for the counts of the number of paths. Using this algorithm we extended and then analysed series for SAWs spanning the square lattice and self-avoiding polygons (SAPs) crossing the square lattice. These are known to also grow as $λ_S^{L^2}.$ The sub-dominant term $λ^b$ is found to be the same as for SAWs crossing the square, while the exponent $g = 1.75\pm 0.01$ for spanning SAWs and $g = -0.500 \pm 0.005$ for SAPs. We have also studied the analogous problems on the hexagonal lattice, and generated series for a number of geometries. In particular, we study SAWs and SAPs crossing rhomboidal, triangular and square domains on the hexagonal lattice, as well as SAWs spanning a rhombus. We estimate that the analogous growth constant $λ_H=1.38724951 \pm 0.00000005,$ so an even more precise estimate than found for the square lattice. We also give estimates of the sub-dominant terms.
