Localization of a Two-Dimensional Random Walk with an Attractive Path Interaction
Localization of a Two-Dimensional Random Walk with an Attractive Path Interaction
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具有吸引路径交互作用的二维随机游走的定位
DOI：
10.1214/aop/1176988734
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发表时间：
1994
影响因子：
2.3
通讯作者：
E. Bolthausen
中科院分区：
文献类型：
--
作者：
E. Bolthausen
我们考虑一个普通的，对称的，连续时间的随机漫步在二维晶格$\mathbb{Z}^2$上。步行的分布由密度转换，该密度以指数方式折扣到时间$T$访问的点的数量。这引入了路径的自吸引相互作用。研究了$T \rightarrow \infty$的渐近性态。结果证明位移是渐近的$T^{1/4}$级。证明结果的主要技术是对大偏差概率的精细分析。对高维也作了部分讨论。
We consider an ordinary, symmetric, continuous-time random walk on the two-dimensional lattice $\mathbb{Z}^2$. The distribution of the walk is transformed by a density which discounts exponentially the number of points visited up to time $T$. This introduces a self-attracting interaction of the paths. We study the asymptotic behavior for $T \rightarrow \infty$. It turns out that the displacement is asymptotically of order $T^{1/4}$. The main technique for proving the result is a refined analysis of large deviation probabilities. A partial discussion is given also for higher dimensions.