Crossings states and sets of states in P\'olya random walks

TL;DR

This paper analyzes the distribution and expectations of state-crossings in Pólya random walks in two and higher dimensions, revealing that the expected number of undirected crossings is always 1 for non-zero states.

Contribution

It establishes that the expected number of undirected state-crossings is 1 for any non-zero state in Pólya random walks and extends these results to higher dimensions and bounded areas.

Findings

Expected number of undirected crossings is 1 for any non-zero state in 2D.

Results are extended to d-dimensional random walks in bounded regions.

Provides detailed distribution and expectation results for state-crossings.

Abstract

We consider the P\'olya random walk in $\mathbb{Z}^2$. The paper establishes a number of results for the distributions and expectations of the number of usual (undirected) and specifically defined in the paper up- and down-directed state-crossings and different sets of states crossings. One of the most important results of this paper is that the expected number of undirected state-crossings $\mathbf{n}$ is equal to 1 for any state $\mathbf{n}\in\mathbb{Z}^2\setminus\{\mathbf{0}\}$. As well, the results of the paper are extended to $d$-dimensional random walks, $d\geq2$, in bounded areas.