Multidimensional Random Walk

Abstract

The simple symmetric random walk on the integers readily extends to that of a simple symmetric random walk on the k-dimensional integer lattice, in which at each step the random walk moves with equal probability to one of its 2k neighboring states on the lattice. A celebrated theorem of Pólya provides a role for dimension k in distinguishing between recurrent and transient properties of the random walk. Namely, it is shown by combinatorial methods that the simple symmetric random walk on the k-dimensional integer lattice ${\mathbb Z}^k$ is recurrent for k = 1, 2 and transient for k ≥ 3.