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

Nina Holden; Pu Yu

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arXiv · 2603.06528

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

Nina Holden·Pu Yu

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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.

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Nina Holden, Pu Yu. 2026-03-06. Circle packing and Riemann uniformization of random planar maps in an ergodic scale-free environment. https://arxiv.org/abs/2603.06528

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