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

TL;DR

This paper investigates the return probability and first return time distribution of a random walker in 2D and 3D continuum with uniformly distributed step lengths, comparing it to fixed step-length walks.

Contribution

It introduces analysis of return probabilities and first return times for random walks with uniformly distributed step lengths in continuous space, a novel extension beyond fixed step-length models.

Findings

Derived the probability distribution of end-to-end distances.

Calculated the probability of return within a finite circular zone.

Analyzed the distribution of first return times.

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.