The circle packing and Riemann uniformization embedding of the tree-weighted planar maps converges to Liouville quantum gravity
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Holden, Nina, Yu, Pu
Institution: New York University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220992121