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PreprintCC BY-SA 4.0arXiv

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 √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 SLE8. 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 √2-LQG by the first author.

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Cite

BibTeXRIS
BibTeX
@misc{holden2026circle,
  title = {{The circle packing and Riemann uniformization embedding of the tree-weighted planar maps converges to Liouville quantum gravity}},
  author = {Nina Holden and Pu Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10505},
  eprint = {2610.10505},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10505},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - The circle packing and Riemann uniformization embedding of the tree-weighted planar maps converges to Liouville quantum gravity
AU  - Nina Holden
AU  - Pu Yu
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10505
UR  - https://arxiv.org/abs/2610.10505
AB  - We prove that in the disk, sphere, whole-plane topology, spanning tree weighted planar maps converge to √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 √2-LQG by the first author.
ER  -