随机游动克服一切困难到达四分之一平面的另一边
Random walks reaching against all odds the other side of the quarter plane
摘要：
对于四分之一平面上具有最近邻跃迁的齐次随机游动，从某个状态$(i_0,j_0)$开始，我们研究了游动在到达水平轴之前到达垂直轴的事件。我们推导出此事件的概率的精确表达式，并推导出当$i_0$变大时，事件变得非常不可能的情况下的渐近表达式。精确的表达式来自于边值问题的解，并且是涉及共形胶合函数的积分。从这个积分的渐近估计的渐近表达式如下。我们的研究结果发现核小体移位模型，选举模型和不对称排它过程中的应用。
英文摘要：
For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state $(i_0,j_0)$, we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive an exact expression for the probability of this event, and derive an asymptotic expression for the case when $i_0$ becomes large, a situation in which the event becomes highly unlikely. The exact expression follows from the solution of a boundary value problem and is in terms of an integral that involves a conformal gluing function. The asymptotic expression follows from the asymptotic evaluation of this integral. Our results find applications in a model for nucleosome shifting, the voter model and the asymmetric exclusion process.
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