Hamiltonian Paths and Self-Avoiding Walks of Lesser Length on Various Surfaces: A Monte Carlo Estimate

Abstract

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 plane squares and rectangles, which is based on the Rosenbluth-Rosenbluth chain-generationprocedure. Starting from a rectangle having m lines and n columns, and thus mXn 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 o, where o is the fraction of available lattice sites on the surface which are occupied by the N-step self-avoinding walk. The variation of the ratio Z NC /Z N with m and n has been estimated for cyclindrical and conical surfaces. Finally, an effective coordination number q eff has been introduced for finite surfaces, and its variatiion with o studied. A cylindrical surface displaying m inscribed circles and four radii. For an N-step self-avoiding walk (N<m/2), the m circles should be divided in m-2N central circles and 2N marginal circles.