论文标题
一维简单随机步行的定位在幂律更新障碍之间
Localization of a one-dimensional simple random walk among power-law renewal obstacles
论文作者
论文摘要
我们考虑一个一维简单的随机步行，被淬灭的软障碍物杀死。障碍物的位置是根据具有幂律增量分布的更新过程来提取的。在先前的工作中，我们计算了猝灭生存概率的大渐近造物。在目前的工作中，我们通过描述为生存的随机行走行为的行为来继续我们的研究。我们证明，步行迅速达到了一个独特的时间依赖于时间的最佳差距，该差距没有障碍，并在那里进行了本地化。我们实际上建立了二分法。如果续签尾部指数小于一个指数，那么步行将击中最佳间隙，并将其剩余的所有时间都花在内部，直至有限的访问到缝隙的底部。 如果续签尾部指数大于一个指数，那么随机步行将大部分时间都花在最佳差距内，但还可以在其长度和基数上提供匹配的上和下限。我们的关键工具包括对生存概率的Markov更新解释以及障碍环境的各种比较论点。我们的结果也可以根据多个排斥界面中的定向聚合物的定位属性来改写。
We consider a one-dimensional simple random walk killed by quenched soft obstacles. The position of the obstacles is drawn according to a renewal process with a power-law increment distribution. In a previous work, we computed the large-time asymptotics of the quenched survival probability. In the present work we continue our study by describing the behaviour of the random walk conditioned to survive. We prove that with large probability, the walk quickly reaches a unique time-dependent optimal gap that is free from obstacle and gets localized there. We actually establish a dichotomy. If the renewal tail exponent is smaller than one then the walk hits the optimal gap and spends all of its remaining time inside, up to finitely many visits to the bottom of the gap. If the renewal tail exponent is larger than one then the random walk spends most of its time inside of the optimal gap but also performs short outward excursions, for which we provide matching upper and lower bounds on their length and cardinality. Our key tools include a Markov renewal interpretation of the survival probability as well as various comparison arguments for obstacle environments. Our results may also be rephrased in terms of localization properties for a directed polymer among multiple repulsive interfaces.
