The Recurrence and Transience of Random Walks

Abstract

This article reviews the research history and application fields of random walks and abstracts random walks into a time-homogeneous Markov chain to study their recurrent and transient properties. For one-dimensional and two-dimensional random walks, the likelihood of returning in  steps and the probability of returning for the first time in  steps of each state are first introduced, along with their relationship. Then the Stirling's formula is given, which is utilized to estimate the probability of returning in n steps, and the convergence and divergence of infinite series is used to prove that random walk in one dimension is recurrent. Random walk in two dimension has similar properties, which is a bit more complicated by using the relation between polynomials. Higher dimensional random walks need to consider special states first, and then generalize them to other states. Finally, this paper concluded that one-dimensional and two-dimensional random walks are recurrent, and random walks with dimensions higher than two are transient.