[论文解读] Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense

Ewain Gwynne, Nina Holden+1 more|arXiv (Cornell University)|May 16, 2019

Stochastic processes and statistical mechanics参考文献 31被引用 6

一句话总结

该论文首次建立了带临界位点渗滤的随机三角剖分在度量和Peanosphere意义下联合收敛于带SLE$_6$的$√{8/3}$-Liouville量子重力（LQG）。通过耦合度量与基于树的编码方法，作者证明了所有渗滤界面集合在Gromov–Hausdorff–Prokhorov–uniform（GHPU）拓扑下收敛于CLE$_6$，同时其对应的随机游走编码收敛于相关布朗运动，从而证实了两种缩放极限下的同步收敛。

ABSTRACT

Recent works have shown that random triangulations decorated by critical ($p=1/2$) Bernoulli site percolation converge in the scaling limit to a $\sqrt{8/3}$-Liouville quantum gravity (LQG) surface (equivalently, a Brownian surface) decorated by SLE$_6$ in two different ways: 1. The triangulation, viewed as a curve-decorated metric measure space equipped with its graph distance, the counting measure on vertices, and a single percolation interface converges with respect to a version of the Gromov-Hausdorff topology. 2. There is a bijective encoding of the site-percolated triangulation by means of a two-dimensional random walk, and this walk converges to the correlated two-dimensional Brownian motion which encodes SLE$_6$-decorated $\sqrt{8/3}$-LQG via the mating-of-trees theorem of Duplantier-Miller-Sheffield (2014); this is sometimes called $ extit{peanosphere convergence}$. We prove that one in fact has $ extit{joint}$ convergence in both of these two senses simultaneously. We also improve the metric convergence result by showing that the map decorated by the full collection of percolation interfaces (rather than just a single interface) converges to $\sqrt{8/3}$-LQG decorated by CLE$_6$ in the metric space sense. This is the first work to prove simultaneous convergence of any random planar map model in the metric and peanosphere senses. Moreover, this work is an important step in an ongoing program to prove that random triangulations embedded into $\mathbb C$ via the so-called $ extit{Cardy embedding}$ converge to $\sqrt{8/3}$-LQG.

研究动机与目标

• 建立带临界位点渗滤的随机三角剖分在度量和Peanosphere意义下的同步收敛。
• 将度量收敛性从单个渗滤界面扩展至整个界面集合，表明其收敛于CLE$_6$-装饰的$√{8/3}$-LQG。
• 为证明随机三角剖分的Cardy嵌入在连续极限下收敛于共形嵌入奠定基础。
• 证明联合缩放极限能够同时捕捉几何（度量）与组合（树编码）结构。

提出的方法

• 利用度量与基于树的编码之间的耦合，同时控制Gromov–Hausdorff–Prokhorov–uniform（GHPU）拓扑与Peanosphere收敛性。
• 采用树的配对双射，通过二维随机游走编码位点渗滤的三角剖分，其收敛于相关布朗运动。
• 应用Gromov–Hausdorff–Prokhorov–uniform（GHPU）拓扑，证明带曲线装饰的度量测度空间的收敛性。
• 使用Schaeffer双射及其推广，将图距离与树标签关联，从而控制度量结构。
• 应用Duplantier–Miller–Sheffield的树配对定理，将随机游走编码与SLE$_6$-装饰的$√{8/3}$-LQG联系起来。
• 采用重根不变性与环集技术，耦合多个空间填充探索过程，证明交叉事件的联合收敛性。

实验结果

研究问题

• RQ1 带临界位点渗滤的随机三角剖分能否在度量与Peanosphere意义下联合收敛于带SLE$_6$的$√{8/3}$-LQG？
• RQ2 整个渗滤界面集合是否在度量空间意义下收敛于CLE$_6$，而不仅限于单个界面？
• RQ3 能否在缩放极限中联合耦合来自不同根边的多个空间填充探索过程，以保持拓扑与概率结构？
• RQ4 随机游走编码的收敛性如何与底层几何结构及曲线装饰的度量结构的收敛性相关联？
• RQ5 联合收敛是否可用于证明Cardy嵌入在连续极限下收敛于共形嵌入？

主要发现

• 该论文首次实现了任何随机平面图模型在度量与Peanosphere意义下的联合收敛。
• 整个渗滤界面集合在GHPU拓扑下收敛于CLE$_6$-装饰的$√{8/3}$-LQG，扩展了以往仅考虑单个界面的研究结果。
• 渗滤三角剖分的随机游走编码收敛于相关二维布朗运动，证实了Peanosphere收敛性。
• 证明了来自不同根边的多个空间填充探索过程的联合收敛性，从而实现了交叉事件向其连续类比的收敛。
• Cardy嵌入向共形嵌入的收敛性已在[HS19]中建立，其基础正是本文所证明的联合缩放极限。
• 度量与编码过程的耦合确保了在连续极限中几何与组合结构均被保留。