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

We prove that embedded infinite plane triangulations 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.

💡 Research Summary

The paper investigates the large‑scale relationship between two canonical discrete conformal embeddings of infinite random planar triangulations: the Koebe‑Andreev‑Thurston circle packing and the Riemann uniformization embedding. The setting is an “ergodic scale‑free environment,” a probabilistic framework introduced by Gwynne, Miller and Sheffield (GMS) in which random walks on certain cell configurations converge, in the quenched sense, to planar Brownian motion. This framework naturally arises in the study of random planar maps and Liouville quantum gravity (LQG).

The authors first formalize a “cell configuration” H as a locally finite collection of compact connected cells covering the complex plane, together with an associated planar map M (the adjacency graph of the cells) and positive conductances on edges. Several structural hypotheses are imposed:

1. Translation‑invariant modulo scaling – after random translation and rescaling, the law of H is unchanged.
2. Ergodicity modulo scaling – any translation‑ and scaling‑invariant observable is almost surely constant.
3. Macroscopic line connectivity – for any ε>0, beyond a random radius R, every horizontal or vertical line segment of length at least εr intersecting the ball B(0,r) induces a connected subgraph of cells.
4. Almost‑planarity – there exist points {z_H} and simple curves {γ_e} that give a planar embedding of the associated map M, and the Hausdorff distance between each cell (or edge) and its embedding decays like o(r) as the observation radius r→∞.

A moment condition is also required: the fourth moment of the product of a typical cell’s diameter, area, and degree must be finite.

Under these assumptions the paper proves two main theorems.

Theorem 1.7 (Circle‑packing closeness). The random cell configuration H is almost surely circle‑packing parabolic (its packing fills the whole plane). Moreover, there exists a deterministic 2×2 matrix A with det A = 1 such that, after applying A to the Euclidean centroid c(H) of each cell, the distance to the center o(H) of the corresponding packing disk tends to zero uniformly on the ball B(0,r) when divided by r. If H is additionally rotation‑invariant (for some angle not equal to π), A can be taken as the identity.

Theorem 1.8 (Riemann‑uniformization closeness). The Riemann surface M(H) obtained by gluing equilateral triangles along the faces of the associated map is almost surely parabolic. There exists a deterministic matrix A (det A = 1) and a conformal map φ₀ : M(H)→ℂ such that, after applying A to the Euclidean centroid of each cell, the distance to φ₀(v_H) (the image of the corresponding vertex) is o(r) uniformly on B(0,r).

The proofs combine several sophisticated tools:

• Ring Lemma and three‑circle theorem – provide uniform control of radii in a circle packing, ensuring that local geometry does not degenerate.
• Dubé‑Jko conductances – a specific choice of edge weights derived from the geometry of the packing; the authors adapt GMS’s quenched invariance principle to these conductances, establishing convergence of the random walk on H (and on the packing) to Brownian motion.
• Vertex extremal length and macroscopic circles – a discrete extremal length argument shows the existence of large circles whose radii grow linearly with the observation scale, which is crucial for the linear‑scale comparison.
• Regularity estimates for the Riemann surface – analytic bounds (e.g., Harnack inequality, gradient estimates) guarantee that the uniformization map does not distort distances excessively.
• Harmonic corrector construction – a function correcting the discrepancy between the discrete centroid and the continuous uniformization coordinate. Sublinearity of the corrector (its growth is o(r)) is proved via a multiscale argument, yielding the desired linear‑scale convergence.

The paper also discusses extensions. The macroscopic line‑connectivity condition can be weakened, the results extend from triangulations to more general planar maps, and analogous statements hold for p‑angulations (maps whose faces have p edges). These extensions broaden the applicability of the main theorems to a wide class of random planar structures.

In summary, the work establishes that, in an ergodic scale‑free setting with mild moment and connectivity assumptions, the two most natural discrete conformal embeddings of a random infinite triangulation are asymptotically indistinguishable on large scales. This provides a rigorous bridge between discrete random geometry (circle packings, Tutte embeddings, etc.) and continuous conformal geometry (Riemann uniformization, LQG), reinforcing the conjectured universality of Liouville quantum gravity as the scaling limit of many random planar map models.