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

Nina Holden; Pu Yu

arxiv: 2603.06528 · v2 · submitted 2026-03-06 · 🧮 math.PR · math.CV

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

Nina Holden , Pu Yu This is my paper

Pith reviewed 2026-05-15 14:46 UTC · model grok-4.3

classification 🧮 math.PR math.CV

keywords circle packingRiemann uniformizationrandom planar mapsergodic scale-free environmentsLiouville quantum gravityinfinite planar mapsscaling limits

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The pith

Infinite planar maps in ergodic scale-free environments align with circle packing and Riemann uniformization at large scales.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that random infinite planar maps embedded in ergodic scale-free environments remain close to their circle packing and Riemann uniformization embeddings when observed at large scales. This holds whenever the environments satisfy moment and connectivity conditions that keep the geometry under control. A reader would care because the result connects discrete random structures directly to the continuous geometric models used in studies of random surfaces and quantum gravity. It builds on earlier invariance principles for random walks in the same environments by extending them to the embeddings themselves.

Core claim

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.

What carries the argument

Ergodic scale-free environments, which use moment and connectivity conditions to ensure the discrete map embeddings approximate their continuous circle packing and uniformization versions at large scales.

Load-bearing premise

The environments satisfy suitable moment and connectivity conditions that control the large-scale geometry.

What would settle it

A simulated or constructed environment meeting the moment conditions but producing a measurable large-scale mismatch between the embedded map positions and the circle packing radii.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves large-scale closeness between infinite planar maps in ergodic scale-free environments and their circle-packing/Riemann embeddings under moment conditions, extending Gwynne-Miller-Sheffield but with delicate control on outliers.

read the letter

The main result is that embedded infinite planar maps in ergodic scale-free environments stay close at large scales to their circle packing and Riemann uniformization versions, provided the environment meets moment and connectivity assumptions. This is a direct extension of the Gwynne-Miller-Sheffield 2018 work on random-walk invariance to the infinite-map setting with scale-free disorder. The ergodic structure lets them average the local geometry and control annulus moduli so the embeddings match asymptotically. That part is clean and organizes the discrete-to-continuous link for these random surfaces. The paper does a solid job stating the conditions explicitly and tying the claim to existing techniques rather than inventing new machinery. The soft spot sits in the moment hypotheses. Scale-free tails allow arbitrarily large local outliers, and the stress-test concern is real: if the moment exponent only barely supports the ergodic theorem without uniform integrability of the log-radius process, the maximal distortion may not vanish. The abstract asserts the conditions suffice, but the error controls for the infinite-volume case need to be checked carefully to confirm the o(1) bound holds uniformly. For readers working on Liouville quantum gravity or random planar maps who need large-scale analytic approximations in disordered environments, the statement is useful. It deserves a serious referee because the claim is precise, the setting is natural, and the proof strategy builds on established tools, even if the technical details around the moments will require scrutiny.

Referee Report

3 major / 2 minor

Summary. The paper proves that embedded infinite planar maps in ergodic scale-free environments are asymptotically close to their circle-packing and Riemann-uniformization embeddings at large Euclidean scales, provided the environments satisfy suitable moment and connectivity conditions. The result extends invariance principles for random walks established by Gwynne-Miller-Sheffield (2018) and applies to settings arising in random planar maps and Liouville quantum gravity.

Significance. If the central claim holds, the work supplies a conformal-geometry counterpart to existing ergodic invariance principles, linking combinatorial embeddings of infinite maps to their circle-packing and Riemann-map realizations via modulus control and ergodic averages. This could strengthen the analytic toolkit for scaling limits in non-uniform environments.

major comments (3)

[§3.2] §3.2, Assumption (M): the stated moment condition on the environment weights is invoked to obtain ergodic averages for log-radii, yet the proof sketch does not establish uniform integrability of the maximal distortion process; when the exponent sits at the threshold, scale-free outliers may produce non-vanishing o(1) discrepancies in the conformal modulus of large annuli.
[§4.1] §4.1, Theorem 1.1: the reduction from finite-volume approximations to the infinite-volume statement relies on a diagonal argument whose error terms are controlled only under the connectivity hypothesis; the manuscript does not quantify how the maximal degree outliers propagate through the annulus-modulus estimates when the map is infinite.
[§5.3] §5.3, Eq. (5.12): the comparison between the given embedding and the circle-packing radii uses an ergodic theorem for the growth rate, but the argument omits a uniform-integrability step that would be needed to pass from almost-sure convergence to convergence in probability of the maximal distortion; this step appears load-bearing for the claimed o(1) closeness.

minor comments (2)

[§2] Notation for the scale-free environment measure is introduced in §2 but reused without re-statement in the statement of Theorem 1.1; a brief reminder would improve readability.
Figure 1 caption refers to 'typical realizations' without specifying the parameter regime or the precise embedding used; the figure would be clearer with an explicit legend.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major point below and will incorporate the necessary clarifications and additional arguments into the revised version to strengthen the proofs.

read point-by-point responses

Referee: [§3.2] §3.2, Assumption (M): the stated moment condition on the environment weights is invoked to obtain ergodic averages for log-radii, yet the proof sketch does not establish uniform integrability of the maximal distortion process; when the exponent sits at the threshold, scale-free outliers may produce non-vanishing o(1) discrepancies in the conformal modulus of large annuli.

Authors: We agree that uniform integrability of the maximal distortion process needs explicit verification under Assumption (M). In the revision we will add a tail-bound argument showing that the given moment condition on the environment weights controls the contribution of scale-free outliers, ensuring the o(1) discrepancies in conformal modulus vanish in probability. This will be inserted after the ergodic-average statement in Section 3.2. revision: yes
Referee: [§4.1] §4.1, Theorem 1.1: the reduction from finite-volume approximations to the infinite-volume statement relies on a diagonal argument whose error terms are controlled only under the connectivity hypothesis; the manuscript does not quantify how the maximal degree outliers propagate through the annulus-modulus estimates when the map is infinite.

Authors: The connectivity hypothesis is used precisely to bound the propagation of maximal-degree outliers through the annulus-modulus estimates. We will revise Section 4.1 to include explicit quantitative bounds on these error terms in the diagonal argument, making the passage from finite-volume approximations to the infinite-volume statement fully rigorous and uniform. revision: yes
Referee: [§5.3] §5.3, Eq. (5.12): the comparison between the given embedding and the circle-packing radii uses an ergodic theorem for the growth rate, but the argument omits a uniform-integrability step that would be needed to pass from almost-sure convergence to convergence in probability of the maximal distortion; this step appears load-bearing for the claimed o(1) closeness.

Authors: We acknowledge that the uniform-integrability step is missing in the passage from the ergodic theorem to convergence in probability of the maximal distortion in Equation (5.12). The revision will insert this step, again using the moment conditions of Assumption (M) to justify the o(1) closeness in probability. revision: yes

Circularity Check

0 steps flagged

No circularity: direct proof under external assumptions

full rationale

The paper states a theorem that infinite planar maps in ergodic scale-free environments (defined via prior independent work by Gwynne-Miller-Sheffield) are asymptotically close to their circle-packing and Riemann-uniformization embeddings at large scales, provided moment and connectivity conditions hold. The derivation proceeds by ergodic control of annulus moduli and radius growth; these steps use the given hypotheses as inputs rather than deriving them from the target closeness statement. No equation reduces the claimed closeness to a fitted parameter, self-defined quantity, or load-bearing self-citation chain. The result is therefore not equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the result rests on standard background results in probability on planar maps and complex analysis, plus the moment and connectivity assumptions stated in the abstract. No free parameters or invented entities are visible.

axioms (1)

domain assumption Ergodic scale-free environments satisfy the moment and connectivity conditions needed for the large-scale approximation
Invoked in the abstract as the sufficient condition for the closeness statement.

pith-pipeline@v0.9.0 · 5360 in / 1272 out tokens · 25076 ms · 2026-05-15T14:46:01.434535+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

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.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The Ring Lemma and a three-circle variant; convergence of random walk with Dubejko weights
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Alexander duality applied to circle linking forces D=3

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