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Hamiltonian paths and self-avoiding walks of lesser length on various surfaces: A Monte Carlo estimate

机译：不同表面上较短长度的哈密顿路径和自我避免的步道：蒙特卡洛估计

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Given some lattice, the number Z(HP) of Hamiltonian paths and also the number Z(N) of N-step shorter self-avoiding walks on the surface of cylinders, cones, tori, and spheres has been Monte Carlo estimated. The procedure is an extension of the technique used in a previous paper for pane squares and rectangles, which is based on the Rosenbluth-Rosenbluth chain-generation procedure. Starting from a rectangle having m lines and n columns, and thus m x n lattice sites, one may obtain cylindrical, conical, toroidal and spherical surfaces through continuous deformations, which respect the topology. Then a correspondence is established between a plane figure of the 'polar' coordinates kind and the topology of the above surfaces. Using this topological equivalence, and thus operating exclusively on the plane 'polar' figure, Monte Carlo simulations show that for given m and n, Z(HP) and Z(N) increase when going from the plane rectangle to the cylinder and then to the cone and the torus. The number Z(NC) of N-step cycles (closed configurations) has also been Monte Carlo estimated. The Monte Carlo results for the surfaces studied here have been condensed in fifth degree polynomials in Phi, where Phi is the fraction of available lattice sites on the surface which are occupied by the N-step self-avoiding walk. The variation of the ratio Z(NC)/Z(N) with m and n has been estimated for cylindrical and conical surfaces, Finally, an effective coordination number q(eff) has been introduced for finite surfaces, and its variation with Phi studied. [References: 23]
机译：给定一些晶格，已经对蒙特卡洛估计了圆柱体，圆锥，圆托和球体表面上的哈密顿路径的Z（HP）数以及N步较短的自避免步道的Z（N）数。该过程是先前论文中用于窗格正方形和矩形的技术的扩展，该技术基于Rosenbluth-Rosenbluth链生成过程。从具有m条线和n列的矩形开始，因此从m x n个晶格位置开始，可以通过遵守拓扑的连续变形来获得圆柱，圆锥，环形和球形表面。然后在“极”坐标类型的平面图形和上述表面的拓扑之间建立对应关系。使用这种拓扑等价关系，并因此仅在平面“极坐标”图上进行操作，蒙特卡洛模拟显示，对于给定的m和n，从平面矩形到圆柱体，然后到，则Z（HP）和Z（N）增加。圆锥和圆环。 N步循环（封闭配置）的数量Z（NC）也已进行了蒙特卡洛估计。此处研究的表面的蒙特卡洛结果已浓缩为Phi的五次多项式，其中Phi是表面上可用的晶格位点的一部分，被N步自回避步态占据。估计了圆柱和圆锥形表面Z（NC）/ Z（N）的比值随m和n的变化，最后，为有限表面引入了有效的配位数q（eff），并研究了其随Phi的变化。 。 [参考：23]

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