Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree
Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree
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DOI：
10.1214/15-aop1030
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发表时间：
2014-07
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影响因子：
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通讯作者：
M. Barlow;D. Croydon;T. Kumagai
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文献类型：
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作者：
M. Barlow;D. Croydon;T. Kumagai
本文的第一个主要结果是（重标化的）二维均匀生成树的定律在元素被测量的空间中是紧的，有根的实树连续嵌入到欧几里德空间中。得到了该空间内的固有度量、测度和后续极限的嵌入的各种性质，特别证明了其固有度量中任意极限的Hausdorff维数几乎一定等于$8/5$。此外，利用紧性结果推导出二维均匀生成树上简单随机游动的退火律在适当的重标度下是紧性的。对于嵌入欧几里得空间的随机实树上的扩散极限过程，导出了详细的转移密度估计。
The first main result of this paper is that the law of the (rescaled) two-dimensional uniform spanning tree is tight in a space whose elements are measured, rooted real trees continuously embedded into Euclidean space. Various properties of the intrinsic metrics, measures and embeddings of the subsequential limits in this space are obtained, with it being proved in particular that the Hausdorff dimension of any limit in its intrinsic metric is almost surely equal to $8/5$. In addition, the tightness result is applied to deduce that the annealed law of the simple random walk on the two-dimensional uniform spanning tree is tight under a suitable rescaling. For the limiting processes, which are diffusions on random real trees embedded into Euclidean space, detailed transition density estimates are derived.