The circle packing and Riemann uniformization embedding of the mated-CRT map converges to Liouville quantum gravity
Abstract
We prove that the mated-CRT map converges to Liouville quantum gravity under circle packing and the Riemann uniformization embedding. The proof is based on our recent work on circle packing and the Riemann uniformization embedding for random planar maps in ergodic-scale free environments, comparison of circle packing in different domains and several existing and new estimates for the mated-CRT map. As an intermediate step, we prove an extension of He and Schramm's result (1996) on convergence of circle packing maps to conformal maps to global uniform topology in several settings. The latter results are of independent interest and will be applied in our subsequent work on the convergence of spanning tree weighted planar maps to √2-LQG under circle packing and Riemann uniformization embedding.
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BibTeX
@misc{holden2026circle,
title = {{The circle packing and Riemann uniformization embedding of the mated-CRT map converges to Liouville quantum gravity}},
author = {Nina Holden and Pu Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10480},
eprint = {2610.10480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10480},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - The circle packing and Riemann uniformization embedding of the mated-CRT map converges to Liouville quantum gravity
AU - Nina Holden
AU - Pu Yu
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10480
UR - https://arxiv.org/abs/2610.10480
AB - We prove that the mated-CRT map converges to Liouville quantum gravity under circle packing and the Riemann uniformization embedding. The proof is based on our recent work on circle packing and the Riemann uniformization embedding for random planar maps in ergodic-scale free environments, comparison of circle packing in different domains and several existing and new estimates for the mated-CRT map. As an intermediate step, we prove an extension of He and Schramm's result (1996) on convergence of circle packing maps to conformal maps to global uniform topology in several settings. The latter results are of independent interest and will be applied in our subsequent work on the convergence of spanning tree weighted planar maps to √2-LQG under circle packing and Riemann uniformization embedding.
ER -