Exploiting the coherent medium approximation, I investigate a random walk on objects distributed randomly in continuous space when jump rate depends distance between two adjacent objects. In one dimension, it is shown that decays exponentially long limit, non-diffusive to diffusive transition occurs as density of sites increased. three dimensions, exists has super Gaussian decay.

The loop-erased random walk (LERW) in three dimensions is obtained by erasing loops chronologically from simple random walk. In this paper we show the existence of the two-sided LERW which can be considered as the distribution of the LERW as seen by a point in the “middle” of the path.