生成树装饰平面图中的有向距离：精确指数、标度极限与普适性
Directed distances in spanning-tree-decorated planar maps: exact exponent, scaling limit and universality
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中文总结 AI 辅助
本文定义生成树装饰平面图的自然定向，研究有向距离，证明其典型阶为n^{1/3}，Busemann函数标度收敛至3/2-稳定Lévy过程，并推广至√2-LQG普适类，给出有向度量的标度维数。
中文摘要 AI 辅助
我们定义了一种生成树装饰平面图的自然定向，粗略地说，图中的每条有向边都定向为与生成树的轮廓探索方向一致。我们研究相对于该定向的有向距离（最短有向路径的长度）。我们构造了Busemann函数，它度量了在均匀无限生成树装饰图中沿自然界面的到无穷远的有向距离。我们证明了该Busemann函数在适当重新标度后在分布上收敛到一个3/2-稳定Lévy过程。我们还证明了在具有n条边的均匀生成树装饰图中，有向距离通常为n^{1/3}阶。利用强耦合论证，我们推导出在√2-刘维尔量子引力（LQG）普适类中的其他随机平面图（包括均匀迷宫系统和γ=√2的配对CRT图）的有向距离的类似结论。这些结果给出了假设的√2-LQG度量的有向版本的标度维数。我们的证明策略受到Borga和Gwynne（2025）关于双极定向三角剖分中有向距离的工作的启发。
英文摘要
We define a natural orientation on a spanning-tree-decorated planar map whereby, roughly speaking, each directed edge in the map is oriented to match the direction of the contour exploration of the spanning tree. We study directed distances (lengths of shortest directed paths) with respect to this orientation. We construct the Busemann function which measures directed distances to $\infty$ along a natural interface in the uniform infinite spanning-tree-decorated map. We show that this Busemann function, re-scaled appropriately, converges in law to a $3/2$-stable Lévy process. We also show that in a uniform spanning-tree-decorated map with $n$ edges, directed distances are typically of order $n^{1/3}$. Using a strong coupling argument, we deduce analogous statements for directed distances in other random planar maps in the $\sqrt 2$-Liouville quantum gravity (LQG) universality class, including uniform meandric systems and mated-CRT maps for $γ=\sqrt 2$. These results give the scaling dimension for a hypothetical directed version of the $\sqrt 2$-LQG metric. Our proof strategy is inspired by work of Borga and Gwynne (2025) on directed distances in bipolar-oriented triangulations.
发表机构
- MIT（麻省理工学院）
- University of Chicago（芝加哥大学）
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