Scaling limit for random walk on the range of random walk in four dimensions
Scaling limit for random walk on the range of random walk in four dimensions
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DOI：
10.1214/22-aihp1243
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发表时间：
2021-04
期刊：
影响因子：
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通讯作者：
D. Croydon;D. Shiraishi
中科院分区：
文献类型：
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作者：
D. Croydon;D. Shiraishi
我们建立的标度极限的随机游动的状态空间是一个简单的随机游动的范围内的四维整数格。这些涉及的渐近行为的图形距离原点和空间位置的随机游走的问题。极限过程类似于模型的高维版本，但需要在标度因子中加入额外的对数项才能看到这些。证明适用于最近开发的机械有关的电阻度量空间和随机过程的缩放，与关键的输入是自然缩放声明的随机行走的不变测度，相关的有效电阻度量，图形距离，和切割时间为基础的简单随机行走。
We establish scaling limits for the random walk whose state space is the range of a simple random walk on the four-dimensional integer lattice. These concern the asymptotic behaviour of the graph distance from the origin and the spatial location of the random walk in question. The limiting processes are the analogues of those for higher-dimensional versions of the model, but additional logarithmic terms in the scaling factors are needed to see these. The proof applies recently developed machinery relating the scaling of resistance metric spaces and stochastic processes, with key inputs being natural scaling statements for the random walk’s invariant measure, the associated effective resistance metric, the graph distance, and the cut times for the underlying simple random walk.