Thick points of random walk and the Gaussian free field
Thick points of random walk and the Gaussian free field
复制标题
DOI：
10.1214/20-ejp433
复制
发表时间：
2018-09
影响因子：
1.4
通讯作者：
A. Jego
中科院分区：
文献类型：
--
作者：
A. Jego
我们考虑随机游走的厚点，即局部时间是最大值的一小部分的点。在二维情形下，我们回答了Dembo，Peres，罗森和Zeitouni的一个问题，并计算了平面随机游动的厚点个数，假设增量是对称的，且具有有限的二阶矩.证明提供了一个流线型的参数的基础上连接到高斯自由场和工程在一个非常一般的设置，包括等径图。在高维中，我们研究了厚点的标度极限。特别是，我们表明，重新缩放的厚点的数量收敛到一个非退化的随机变量和当地时间的中心最大值收敛到一个随机移动的Gumbel分布。
We consider the thick points of random walk, i.e. points where the local time is a fraction of the maximum. In two dimensions, we answer a question of Dembo, Peres, Rosen and Zeitouni and compute the number of thick points of planar random walk, assuming that the increments are symmetric and have a finite moment of order two. The proof provides a streamlined argument based on the connection to the Gaussian free field and works in a very general setting including isoradial graphs. In higher dimensions, we study the scaling limit of the thick points. In particular, we show that the rescaled number of thick points converges to a nondegenerate random variable and that the centred maximum of the local times converges to a randomly shifted Gumbel distribution.