圆堆积与黎曼均匀化嵌入下树加权平面地图收敛于刘维尔量子引力
The circle packing and Riemann uniformization embedding of the tree-weighted planar maps converges to Liouville quantum gravity
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中文总结 AI 辅助
本文证明树加权平面地图在圆堆积和黎曼均匀化嵌入下收敛于$\sqrt{2}$-LQG，并附带证明自然路径收敛于SLE$_8$。
中文摘要 AI 辅助
我们证明了在圆盘、球面、全平面拓扑中，生成树加权平面地图在圆堆积和黎曼均匀化嵌入下，当地图的面数趋于无穷时，收敛于$\sqrt{2}$-刘维尔量子引力圆盘、球面或锥。作为副产品，我们还证明了嵌入的树加权平面地图上面上的自然路径收敛于SLE$_8$。该证明基于我们早先在遍历无标度环境中随机平面地图的圆堆积和黎曼均匀化嵌入的工作、配套论文中不同域中圆堆积的比较，以及第一作者关于树加权平面地图收敛于$\sqrt{2}$-LQG的结果。
英文摘要
We prove that in the disk, sphere, whole-plane topology, spanning tree weighted planar maps converge to $\sqrt{2}$-Liouville quantum gravity disk, sphere or cone under circle packing and Riemann uniformization embedding as the number of faces of the map goes to infinity. As a byproduct, we also prove that the natural path on faces of the embedded tree-weighted planar maps converge to SLE$_8$. The proof is based on our earlier work on circle packing and Riemann uniformization embedding for random planar maps in ergodic scale-free environments, comparisons of circle packings in different domains in a companion paper, and the convergence of tree-weighted planar maps to $\sqrt{2}$-LQG by the first author.
发表机构
- Courant Institute, New York University（纽约大学柯朗研究所）
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