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arXiv · 2610.10505

The circle packing and Riemann uniformization embedding of the tree-weighted planar maps converges to Liouville quantum gravity

Nina Holden·Pu Yu
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Abstract

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.

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Nina Holden, Pu Yu. 2026-10-07. The circle packing and Riemann uniformization embedding of the tree-weighted planar maps converges to Liouville quantum gravity. https://arxiv.org/abs/2610.10505

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