Random walk, Pólya’s Recurrence Theorem, Brownian motion
Abstract
The phenomena of random walks are omniscient in nature. in this study, the author conducts an in-depth exploration of the properties of one, two, and three-dimensional random walks, emphasizing their interconnections. Through simulations of random walks across each dimension, the author carefully analyzes the displacement distribution and the mean squared displacement to classify the walks as either recurrent or transient. The results validate Pólya’s Recurrence Theorem, demonstrating that random walks in one and two dimensions are recurrent, meaning that the walker has a high probability of returning to the origin. In contrast, random walks in three dimensions tend to be transient, where the walker is less likely to revisit the starting point. These findings are essential in the broader context of understanding Brownian motion, particularly in nanoconfined environments like DNA, where random motion significantly impacts molecular behavior. This study helps provide insights into how spatial constraints influence random movements at the nanoscale level.
