Beyond Polya’S Random Walk Theorem

Abstract

A random walk can be regarded as a probability model depicting some degree of randomness, which has lots of interdisciplinary applications in physics, biochemistry and computer science. In this paper, the recurrence classifications of five different random walk models are presented along with their relevant studies. In order to explore the essential reasons leading to the qualitative change of simple random walks’ recurrence property, the classification results of the five simple random walk variants are horizontally discussed. As a result, a positive bias is found to be the denominator shared by random walk variants whose recurrence classifications are different from that of simple random walks. A limited walking direction is also found useful in reversing the recurrence result. Besides answering the qualitative change question, this paper is also dedicated to provide a summary of the recurrence properties of different current random walk models, in order to help researchers in related fields to quickly get the picture of some random walk models.