Title: Recurrence and Polya number of general one-dimensional random walks Show Chinese title

Title: 一维随机游走的递归性和Polya数

Abstract: The recurrence properties of random walks can be characterized by P\'{o}lya 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 P\'{o}lya number $P$ of this model and find a simple expression for $P$ as, $P=1-\Delta$, where $\Delta$ is the absolute difference of $l$ and $r$ ($\Delta=|l-r|$). 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 $r$. Show Chinese abstract

Abstract: 随机游走的遍历性质可以通过波利亚数来表征，即行走者至少一次返回原点的概率。 在本文中，我们考虑线上的广义一维随机游走的遍历性质，其中在每个时间步，行走者可以以概率$l$向左移动或以概率$r$向右移动，或者以概率$o$保持在原位置（$l+r+o=1$）。 我们计算该模型的 Pólya 数$P$，并发现$P$的一个简单表达式为$P=1-\Delta$，其中$\Delta$是$l$和$r$的绝对差 ($\Delta=|l-r|$)。 我们通过创造性的望远镜方法证明了这个严格的表达式，我们的结果表明，当左移概率$l$等于右移概率$r$时，该随机游走是常返的。