The interpolation between random walk and self-avoiding walk by avoiding marked sites

Abstract

The self-avoiding walk (SAW) on a regular lattice is one of the most important and classic problems in statistical mechanics with major applications in polymer chemistry. Random walk is an exactly solved problem while SAW is an open problem so far. We interpolate these two limits in 1D and 2D by considering a model in which the walker marks certain sites in time and does not visit them again. We study two variants: (a) the walker marks sites at every k time-steps, in this case, it is possible to enumerate all possible paths of a given length. (b) The walker marks sites with a certain probability p. For k = 1 or p = 1, the walk reduces to the usual SAW. We compute the average trapping time and distance covered by a walker as a function of the number of steps and radius of gyration in these cases. We observe that 1D deterministic, 1D probabilistic, and 2D deterministic cases are in the same universality class as SAW while 2D probabilistic case shows continuously varying exponents.