Recurrence and Polya number of general one-dimensional random walks

The recurrence properties of random walks can be characterized by Pólya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we consider recurrence properties for a general 1D random walk on a line, in which at each time step the walker can move to the left or right with probabilities $l$ and $r$, or remain at the same position with probability $o$ ($l+r+o=1$). We calculate Pólya number $P$ of this model and find a simple expression for $P$ as, $P=1-Δ$, where $Δ$ is the absolute difference of $l$ and $r$ ($Δ=|l-r|$). We prove this rigorous expression by the method of creative telescoping, and our result suggests that the walk is recurrent if and only if the left-moving probability $l$ equals to the right-moving probability $r$.

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