Return Probability of a Random walker in continuum with uniformly distributed jump-length

Abstract: We are studying the motion of a random walker in two and three dimensional continuum with uniformly distributed jump-length. This is different from conventional Lavy flight. In 2D and 3D continuum, a random walker can move in any direction, with equal probability and with any step-length(l) varying randomly between 0 to 1 with equal probability. A random walker, who starts his journey from a particular point (taken as origin), walks through the region, centered around the origin. Here, in this paper, we studied the probability distribution of end-to-end distances and the probability of return for the first time within a circular zone of finite radius, centered about the initial point, of a random walker. We also studied the same when a random walker moves with constant step-length, unity. The probability distribution of the time of first return (within a specified zone) is also studied.