The principle and applications of random walks in various disciplines

Abstract

Given the precondition that the probability of moving in each direction is identical, the properties (recurrent or transient) of random walks in dimension 1, 2 and 3 are discussed. Random walks in dimension 1 and 2 are verified recurrent while in dimension 3, they are proved transient. Afterward, random walks’ applications in different scientific fields are demonstrated and listed. In finance and biology areas, random walk hypothesis (the fluctuation of stock prices follows random walks and is unpredictable) and Lévy walks (a special class of random walks with move steps follow a power-law distribution) are introduced separately. In addition, their applications in physics (simulating the movement route of a molecule in a liquid or gas) and computer science (network embedding, semi-supervised learning and computer vision) are mentioned briefly. Finally, the current limitations and future development of random walks are considered. These results shed light on the importance of random walks in modern science and pave a path for future implementation of it in multiple fields.