Scaling Argument of Anisotropic Random Walk

In this paper, we analytically discuss the scaling properties of the average square end-to-end distance <R2> for anisotropic random walk in D-dimensional space ( D ≥ 2), and the returning probability Pn(ro) for the walker into a certain neighborhood of the origin. We will not only give the calculating formula for <R2> and Pn (ro), but also point out that if there is a symmetric axis for the distribution of the probability density of a single step displacement, we always obtain <R2⊥n> ～ n, where ⊥ refers to the projections of the displacement perpendicular to each symmetric axes of the walk; in D-dimensional space with D symmetric axes perpendicular to each other, we always have <R2n> ～ n and the random walk will be like a purely random motion; if the number of inter-perpendicular symmetric axis is smaller than the dimensions of the space, we must have <R2n> ～ n2 for very large n and the walk will be like a ballistic motion. It is worth while to point out that unlike the isotropic random walk in one and two dimensions, which is certain to return into the neighborhood of the origin, generally there is only a nonzero probability for the anisotropic random walker in two dimensions to return to the neighborhood.

In this paper, we analytically discuss the scaling properties of the average square end-to-end distance < R-2 > for anisotropic random walk in D-dimensional space (D >= 2), and the returning probability P-n(r(0)) for the walker into a certain neighborhood of the origin. We will not only give the calculating formula for < R-2 > and P-n(r(0)), but also point out that if there is a symmetric axis for the distribution of the probability density of a single step displacement, we always obtain < R-perpendicular to n(2) > similar to n, where perpendicular to refers to the projections of the displacement perpendicular to each symmetric axes of the walk; in D-dimensional space with D symmetric axes perpendicular to each other, we always have < R-n(2)> similar to n and the random walk will be like a purely random motion; if the number of inter-perpendicular symmetric axis is smaller than < R-n(2)> similar to n(2) the dimensions of the space, we must have n for very large n and the walk will be like a ballistic motion. It is worth while to point out that unlike the isotropic random walk in one and two dimensions, which is certain to return into the neighborhood of the origin, generally there is only a nonzero probability for the anisotropic random walker in two dimensions to return to the neighborhood.