Circle packing and Riemann uniformization of random planar maps in an ergodic scale-free environment
Probability
2026-05-08 v2 Complex Variables
Abstract
We prove that embedded infinite planar maps in ergodic scale-free environments are close to their circle packing and Riemann uniformization embedding on a large scale, as long as suitable moment and connectivity conditions are satisfied. Ergodic scale-free environments were earlier considered by Gwynne, Miller and Sheffield (2018) in the context of the invariance principles for random walk, and they arise naturally in the study of random planar maps and Liouville quantum gravity.
Keywords
Cite
@article{arxiv.2603.06528,
title = {Circle packing and Riemann uniformization of random planar maps in an ergodic scale-free environment},
author = {Nina Holden and Pu Yu},
journal= {arXiv preprint arXiv:2603.06528},
year = {2026}
}
Comments
55 pages, 9 figures; Extended from triangulations to general planar maps and removed almost planarity condition from the previous version