Scaling limits of the three-dimensional uniform spanning tree and associated random walk
Scaling limits of the three-dimensional uniform spanning tree and associated random walk
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DOI：
10.1214/21-aop1523
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发表时间：
2020-03
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通讯作者：
Omer Angel;D. Croydon;Saraí Hernández-Torres;D. Shiraishi
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作者：
Omer Angel;D. Croydon;Saraí Hernández-Torres;D. Shiraishi
我们证明了三维均匀生成树（UST）的规律在一个元素被度量、根为真实的树、连续嵌入到欧氏空间的空间中的重标度下是紧的。我们还建立了相关的法律实际上收敛沿着一个特定的标度序列。我们用来建立这些结果的技术被进一步应用于获得各种性质的内在度量和措施的任何限制空间，包括显示，Hausdorff维数这样的是由3/\beta$，其中$\beta\approximat1.624\dots $是三维的增长指数的循环擦除随机游走。此外，我们研究了三维均匀生成树上的随机行走，推导出其行走维数（相对于内在和欧几里得度量）和谱维数，证明了其退火定律在重新缩放下的紧密性，并推导出任何扩散的热核估计作为缩放限制。
We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant laws actually converge along a particular scaling sequence. The techniques that we use to establish these results are further applied to obtain various properties of the intrinsic metric and measure of any limiting space, including showing that the Hausdorff dimension of such is given by $3/\beta$, where $\beta\approx 1.624\dots$ is the growth exponent of three-dimensional loop-erased random walk. Additionally, we study the random walk on the three-dimensional uniform spanning tree, deriving its walk dimension (with respect to both the intrinsic and Euclidean metric) and its spectral dimension, demonstrating the tightness of its annealed law under rescaling, and deducing heat kernel estimates for any diffusion that arises as a scaling limit.