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

Given some lattice, the number Z(HP) of Hamiltonian paths and also the numb er 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 a n extension of the technique used in a previous paper for pane squares and rectangles, which is based on the Rosenbluth-Rosenbluth chain-generation pr ocedure. Starting from a rectangle having m lines and n columns, and thus m x n lattice sites, one may obtain cylindrical, conical, toroidal and spher ical 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 topolog ical equivalence, and thus operating exclusively on the plane 'polar' figur e, Monte Carlo simulations show that for given m and n, Z(HP) and Z(N) incr ease when going from the plane rectangle to the cylinder and then to the co ne 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 surfa ces studied here have been condensed in fifth degree polynomials in Phi, wh ere 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.