论文标题
多个随机步行的碰撞度量的缩放限制
Scaling limit of the collision measures of multiple random walks
论文作者
论文摘要
对于整数$ k \ ge 2 $，令$ s^{（1）}，s^{（2）}，\ dots，s^{（k）} $ be $ k $独立的简单对称随机随机步行$ \ mathbb {z} $。如果至少有两个不同的随机步行，即$ s^{（i）}，s^{（j）} $满足$ s^{（i）_ n = s^{（j）} _ n = z $，则一对$（n，z）$称为碰撞事件。我们表明，在与Donsker定理相同的缩放下，代表这些碰撞事件的随机度量顺序会收敛到$ [0,1] \ Times \ Mathbb {R} $上的非平凡随机度量。此外，可以使用Wiener混乱来表征极限随机度量。该证明是受统计力学方法的启发，特别是是由用于研究随机环境中定向聚合物的分区函数的启发。
For an integer $k\ge 2$, let $S^{(1)}, S^{(2)}, \dots, S^{(k)}$ be $k$ independent simple symmetric random walks on $\mathbb{Z}$. A pair $(n,z)$ is called a collision event if there are at least two distinct random walks, namely, $S^{(i)},S^{(j)}$ satisfying $S^{(i)}_n= S^{(j)}_n=z$. We show that under the same scaling as in Donsker's theorem, the sequence of random measures representing these collision events converges to a non-trivial random measure on $[0,1]\times \mathbb{R}$. Moreover, the limit random measure can be characterized using Wiener chaos. The proof is inspired by methods from statistical mechanics, especially, by a partition function that has been developed for the study of directed polymers in random environments.
