Eventual return probability in multidimensional random walks

Vyacheslav M. Abramov

Published 2016-01-11Version 1

It is well-known that the simple symmetric random walk of dimension 1 or 2 is recurrent, i.e. being started from the origin, it returns to the original point infinitely many times. Simple symmetric random walks of dimension higher than 2 are not recurrent. The present paper derives simple limiting recurrence relations, which enable us to find an approximate value of the eventual return probability for a simple symmetric random walk of an arbitrary dimension $d$. The numerical calculations show that the eventual return probabilities for the simple symmetric random walks in ${Z}^3$ and ${Z}^4$ are close to 1.