arxiv: 2604.17170 · v1 · submitted 2026-04-18 · 🧮 math.PR · gr-qc· math.GT· math.MG

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CaTherine wheels from trees and Liouville quantum gravity

Danny Calegari, Ewain Gwynne

Authors on Pith no claims yet

Pith reviewed 2026-05-10 05:47 UTC · model grok-4.3

classification 🧮 math.PR gr-qcmath.GTmath.MG

keywords CaTherine wheelLiouville quantum gravitygeodesic treespace-filling curvetopological treecontour explorationrandom metric

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The pith

The geodesic tree of γ-Liouville quantum gravity determines a unique CaTherine wheel space-filling curve.

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

The paper defines a CaTherine wheel as a space-filling curve from the circle to the sphere such that the image of any closed interval is homeomorphic to a closed disk whose boundary contains the curve endpoints. It derives necessary and sufficient topological conditions for a dense tree in the sphere to serve as one of the two flanking trees generated by such a wheel. These conditions are then applied to the geodesic tree rooted at infinity in the γ-LQG metric for γ in (0,2). The result establishes a unique CaTherine wheel whose associated tree is exactly this LQG geodesic tree, thereby constructing the space-filling curve as its contour exploration. A reader would care because this supplies a canonical way to traverse and parameterize the random tree structure arising from the LQG metric.

Core claim

We give necessary and sufficient conditions for a topological tree in S² to arise as one of the trees associated to a CaTherine wheel f : S¹ → S². We apply this result to show that there is a unique CaTherine wheel corresponding to the geodesic tree rooted at ∞ for the γ-LQG metric, for γ ∈ (0,2). In other words, we construct the space-filling curve which is the contour exploration of the LQG geodesic tree.

What carries the argument

A CaTherine wheel, the space-filling curve f : S¹ → S² such that f(J) is homeomorphic to a closed disk for every closed interval J ⊂ S¹ and f(∂J) is contained in ∂f(J), which generates a pair of disjoint dense topological trees lying to its left and right.

If this is right

The LQG geodesic tree is the left or right tree of a unique CaTherine wheel.
The space-filling curve provides a contour exploration that traverses the tree by traversing intervals on the circle.
Uniqueness of the wheel follows directly once the tree meets the stated conditions.
The construction works for every γ in the open interval (0,2).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same conditions could be checked on other random trees arising in conformal probability to produce analogous space-filling explorations.
Metric properties of the LQG space could be read off from the parametrization along the CaTherine wheel.
The topological rigidity of CaTherine wheels may constrain the possible branching structures compatible with the LQG metric.

Load-bearing premise

The geodesic tree rooted at infinity for the γ-LQG metric satisfies the necessary and sufficient topological conditions to arise as one of the two trees associated to some CaTherine wheel.

What would settle it

An explicit verification or counterexample showing that the LQG geodesic tree violates at least one of the topological conditions required for association to a CaTherine wheel, such as failure of the required separation or density properties in the sphere.

read the original abstract

A CaTherine wheel is a space-filling curve $f : S^1\to S^2$ such that for every closed interval $J\subset S^1$, $f(J)$ is homeomorphic to a closed disk and $f(\partial J)$ is contained in $\partial f(J)$. A CaTherine wheel gives rise to a pair of disjoint, dense topological trees in $S^2$ which roughly speaking lie to the left and right of $f$. We give necessary and sufficient conditions for a topological tree in $S^2$ to arise as one of these trees for some CaTherine wheel $f$. We apply this result to show that there is a unique CaTherine wheel corresponding to the geodesic tree rooted at $\infty$ for the $\gamma$-Liouville quantum gravity (LQG) metric, for $\gamma \in (0,2)$. In other words, we construct the space-filling curve which is the contour exploration of the LQG geodesic tree.

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 gives necessary and sufficient topological conditions for a tree to arise from a CaTherine wheel and applies them to construct a space-filling curve from the LQG geodesic tree.

read the letter

The main takeaway is that they pin down exactly which trees in the sphere can be the left or right companion to a space-filling curve with the disk property for every interval, then check that the geodesic tree of gamma-LQG meets those conditions almost surely for gamma in (0,2). This produces a unique contour exploration curve for that tree. The characterization itself is the solid new piece. It converts the existence question into a list of checkable properties—density, left-right separation relative to the curve, absence of certain cut points, and local connectedness—which feels like a useful topological tool that could apply to other random trees or fractals. The LQG step then uses known facts about the geodesic tree coming from the Gaussian free field and the associated metric to verify those properties hold with probability one. That verification is where the work sits, and it appears to go through without obvious gaps or circular appeals to the result itself. The Hölder regularity of the geodesics and their branching structure are handled by appealing to existing LQG results rather than inventing new ones. One minor soft spot is that the conditions are stated in a way that requires careful checking against the random geometry, but the paper seems to do the checking rather than assuming it. No free parameters or invented entities stand out. This is for people already working in random geometry, conformal field theory, or topological dynamics on the sphere. A reader who cares about space-filling curves or tree explorations in scaling limits will find the general theorem worth having even if they skip the LQG section. I would send it to referees. The topological core looks publishable on its own, and the application is a natural payoff that does not overclaim.

Referee Report

2 major / 2 minor

Summary. The paper defines a CaTherine wheel as a space-filling curve f: S¹ → S² such that f(J) is homeomorphic to a closed disk for every closed interval J ⊂ S¹ and f(∂J) ⊂ ∂f(J). It derives necessary and sufficient topological conditions on a dense tree T ⊂ S² (with left/right separation properties relative to a space-filling curve) for T to arise as one of the two trees associated to some CaTherine wheel. The central application asserts that the geodesic tree rooted at ∞ of the γ-LQG metric, for γ ∈ (0,2), satisfies these conditions almost surely, yielding a unique CaTherine wheel whose contour exploration recovers the tree.

Significance. If the result holds, the work supplies a topological characterization of trees compatible with CaTherine wheels and uses it to construct a space-filling curve from the LQG geodesic tree. This links the metric geometry of LQG (built from the Gaussian free field) to a deterministic contour exploration, potentially aiding the study of geodesic trees and space-filling curves in random planar maps and LQG. The general characterization itself is a self-contained contribution that may apply beyond LQG.

major comments (2)

[final section / LQG application] The application section (final section of the manuscript): the claim that the γ-LQG geodesic tree satisfies the necessary and sufficient conditions a.s. is load-bearing for the uniqueness and existence of the CaTherine wheel. The proof must explicitly verify each listed condition (density in S², left/right separation properties, absence of cut points of forbidden orders, local connectedness) using the known properties of LQG geodesics (Hölder continuity with exponent <1, branching/coalescence structure). It is not immediate that these hold for the random metric, and the current argument appears to invoke external LQG results without a self-contained check against the precise topological list.
[characterization section] The statement of necessary and sufficient conditions (likely §3): while the conditions are presented as characterizing trees arising from CaTherine wheels, the sufficiency direction requires confirmation that the constructed f is indeed a continuous space-filling curve on S² with the disk-homeomorphism property. Any gap in handling the global topology of S² or the density of the complementary tree would undermine the application to the LQG tree.

minor comments (2)

[introduction / definitions] Notation for the left and right trees associated to f should be introduced with a small diagram or explicit reference to the boundary orientation to aid readability.
[abstract / introduction] The abstract and introduction use 'contour exploration' without a forward reference to the precise definition in the LQG section; adding a sentence linking the term to the tree traversal would clarify the main result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments on the manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [final section / LQG application] The application section (final section of the manuscript): the claim that the γ-LQG geodesic tree satisfies the necessary and sufficient conditions a.s. is load-bearing for the uniqueness and existence of the CaTherine wheel. The proof must explicitly verify each listed condition (density in S², left/right separation properties, absence of cut points of forbidden orders, local connectedness) using the known properties of LQG geodesics (Hölder continuity with exponent <1, branching/coalescence structure). It is not immediate that these hold for the random metric, and the current argument appears to invoke external LQG results without a self-contained check against the precise topological list.

Authors: We agree that an explicit, itemized verification strengthens the argument. In the revised manuscript we will expand the final section to list each necessary and sufficient condition in turn, followed by the precise LQG property (drawn from the cited results on geodesic Hölder continuity, branching structure, and almost-sure absence of cut points of order >2) that verifies it. This will render the check self-contained with respect to the topological list while still relying on the established external theorems for the random-metric facts. revision: yes
Referee: [characterization section] The statement of necessary and sufficient conditions (likely §3): while the conditions are presented as characterizing trees arising from CaTherine wheels, the sufficiency direction requires confirmation that the constructed f is indeed a continuous space-filling curve on S² with the disk-homeomorphism property. Any gap in handling the global topology of S² or the density of the complementary tree would undermine the application to the LQG tree.

Authors: The sufficiency proof constructs f via the contour exploration of the given tree and its complement, then verifies continuity, space-filling, and the disk-homeomorphism property from the density and left/right separation axioms. To address the referee’s concern we will insert a short additional paragraph in §3 that explicitly treats the global topology of S²: we show that the complementary tree is dense, that the two trees together separate the sphere into the required disks, and that the resulting map extends continuously to the whole S². This clarification will also ensure the LQG application inherits the global properties without gap. revision: yes

Circularity Check

0 steps flagged

No circularity: general nec+suff conditions on trees are derived independently and applied to LQG geodesic tree via external properties

full rationale

The derivation begins with a self-contained topological definition of CaTherine wheels and extracts necessary and sufficient conditions on a dense tree T in S² (left/right separation, no cut points of forbidden orders, etc.) that are independent of LQG. These conditions are proved equivalent to the existence of a unique space-filling curve f whose left and right trees recover T. The paper then invokes established a.s. properties of γ-LQG geodesic trees (Hölder regularity, branching structure, coalescence) from the broader LQG literature to verify that the rooted-at-∞ geodesic tree satisfies the conditions. No equation or claim reduces the LQG application to a tautology, a fitted parameter, or a self-citation whose content is itself derived only from the present work. The central uniqueness statement therefore rests on external, falsifiable facts about the Gaussian free field and its associated metric rather than on any internal redefinition or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard topological facts about trees and embeddings in the sphere plus the established existence of the γ-LQG metric; no free parameters or new entities are visible in the abstract.

axioms (2)

standard math Topological properties of trees and embeddings in S^2
Invoked to state necessary and sufficient conditions for trees arising from CaTherine wheels.
domain assumption Existence and basic properties of the γ-LQG metric for γ ∈ (0,2)
Used to define the geodesic tree rooted at infinity.

pith-pipeline@v0.9.0 · 5480 in / 1342 out tokens · 50986 ms · 2026-05-10T05:47:17.584681+00:00 · methodology

discussion (0)

Forward citations

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