Circle packing and Riemann uniformization of random planar maps in an ergodic scale-free environment

TL;DR

This paper demonstrates that infinite planar maps in ergodic scale-free environments can be approximated by circle packing and Riemann uniformization embeddings on large scales under certain conditions.

Contribution

It establishes a connection between ergodic scale-free environments and their geometric embeddings, extending previous work on random planar maps and Liouville quantum gravity.

Findings

Embedded maps are close to their circle packing embeddings on large scales.

Conditions on moments and connectivity ensure the approximation accuracy.

Builds on prior work linking ergodic environments with random walk invariance principles.

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.