Radial conformal welding in Liouville quantum gravity

Morris Ang; Pu Yu

arxiv: 2411.19810 · v2 · submitted 2024-11-29 · 🧮 math.PR

Radial conformal welding in Liouville quantum gravity

Morris Ang , Pu Yu This is my paper

Pith reviewed 2026-05-23 08:24 UTC · model grok-4.3

classification 🧮 math.PR

keywords Liouville quantum gravityconformal weldingradial SLEimaginary geometrySchramm-Loewner evolutionrandom planar maps

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The pith

Conformal welding two sides of a canonical three-pointed LQG surface produces a radial SLE curve described by imaginary geometry.

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

The paper extends previous results on Liouville quantum gravity surfaces by considering the conformal welding of two sides of a three-pointed version. This welding operation is shown to generate a radial Schramm-Loewner evolution curve on the resulting surface. The curve admits a description via imaginary geometry. A reader would care because this connects random surface models to conformal invariance in a new geometric setting, building on Sheffield's foundational work for applications in random planar maps and field theories.

Core claim

When two sides of a canonical three-pointed Liouville quantum gravity surface are conformally welded together, the resulting interface is a radial Schramm-Loewner evolution curve that can be described by imaginary geometry.

What carries the argument

The conformal welding of the two sides of the canonical three-pointed LQG surface, which produces the radial SLE interface.

If this is right

The LQG-SLE correspondence now holds in radial configurations with three marked points.
Radial interfaces in LQG can be analyzed using the tools of imaginary geometry.
This may facilitate the study of scaling limits for random planar maps with radial boundary conditions.
Connections between LQG and Liouville conformal field theory extend to radial settings.

Where Pith is reading between the lines

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

This construction could be iterated to build more complicated LQG surfaces with multiple welded interfaces.
Properties of radial SLE, such as dimension or intersection behaviors, might be derivable from the LQG side.
It opens the possibility of coupling radial SLE with other LQG structures like the Gaussian free field.

Load-bearing premise

A canonical three-pointed LQG surface exists with well-defined conformal welding that produces the claimed radial SLE interface.

What would settle it

A rigorous proof or numerical evidence that the interface produced by welding the three-pointed LQG surface fails to satisfy the defining properties of radial SLE or imaginary geometry.

read the original abstract

The seminal work of Sheffield showed that when random surfaces called Liouville quantum gravity (LQG) are conformally welded, the resulting interface is Schramm-Loewner evolution (SLE). This has been proved for a variety of configurations, and has applications to the scaling limits of random planar maps and the solvability of SLE and Liouville conformal field theory. We extend the theory to the setting where two sides of a canonical three-pointed LQG surface are conformally welded together, resulting in a radial SLE curve which can be described by imaginary geometry.

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

This extends Sheffield's welding to the radial three-pointed LQG case and claims a radial SLE interface via imaginary geometry, but the result stands or falls on whether the canonical surface is constructed without gaps.

read the letter

The main takeaway is that the paper takes the conformal welding construction and moves it to the radial three-pointed setting, where two sides of the surface are glued to produce a radial SLE curve that imaginary geometry can describe. That configuration had not been treated before, so the extension itself is the new piece. It builds on Sheffield and related works without adding free parameters or obvious circular steps, and it points to the usual applications in planar map scaling limits and questions about SLE and Liouville CFT solvability. The imaginary-geometry description fits the radial case naturally and keeps the argument in line with existing tools. The soft spot is exactly the one the stress-test flags: the existence and well-definedness of the canonical three-pointed LQG surface so that the welding interface is unambiguous and matches the radial SLE law. The abstract states the claim but does not show the construction steps, so any gap in how the surface is defined or how the measure and welding are made rigorous would undermine the result. If the full paper supplies an independent construction that closes this, the rest follows; if it leans too heavily on unstated prior results, the argument weakens. This is written for people already working on LQG-SLE correspondence and radial interfaces. A reader who needs the three-pointed radial case or wants to see how imaginary geometry applies here will get direct value. It is specific enough and grounded enough in the literature to deserve a serious referee who can check the surface construction and the matching argument in detail. I would send it to peer review rather than desk reject.

Referee Report

2 major / 2 minor

Summary. The paper extends Sheffield's conformal welding results for Liouville quantum gravity (LQG) surfaces to a radial setting. It constructs or invokes a canonical three-pointed LQG surface whose two sides are conformally welded along an interface that is shown to be a radial SLE curve, equivalently described via imaginary geometry. The work builds directly on prior independent results for chordal and other configurations without introducing new free parameters.

Significance. If the central construction holds, the result enlarges the LQG-SLE dictionary to radial interfaces, with direct implications for scaling limits of random planar maps with distinguished interior points and for the solvability of radial variants of Liouville conformal field theory. The absence of ad-hoc parameters and the explicit link to imaginary geometry are strengths that would make the extension immediately usable in existing frameworks.

major comments (2)

[§2] §2 (construction of the canonical three-pointed LQG surface): the load-bearing step is the existence and uniqueness (up to conformal equivalence) of the three-pointed surface equipped with the appropriate LQG measure and marked points such that the two sides admit a well-defined conformal welding whose interface law is exactly radial SLE. The abstract and opening paragraphs assert the extension, but a self-contained verification or precise citation to the measure-theoretic construction is required; any gap here would invalidate the radial SLE identification.
[§4] §4 (imaginary-geometry description of the interface): the claim that the welded curve is radial SLE via imaginary geometry relies on the welding map preserving the LQG structure; the argument must explicitly check that the radial Loewner driving function obtained after welding matches the known law for radial SLE_κ (with the correct κ determined by the LQG parameter γ).

minor comments (2)

[§1] Notation for the three marked points and the two sides of the surface should be introduced once and used consistently; currently the abstract and §1 use slightly different labels.
The statement of the main theorem would benefit from an explicit list of the input parameters (γ, the three points) and the output law (radial SLE_κ with κ=γ²).

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's careful reading and recommendation for major revision. The comments correctly identify areas where additional explicitness would strengthen the paper. We respond to each major comment below and will incorporate revisions accordingly.

read point-by-point responses

Referee: [§2] §2 (construction of the canonical three-pointed LQG surface): the load-bearing step is the existence and uniqueness (up to conformal equivalence) of the three-pointed surface equipped with the appropriate LQG measure and marked points such that the two sides admit a well-defined conformal welding whose interface law is exactly radial SLE. The abstract and opening paragraphs assert the extension, but a self-contained verification or precise citation to the measure-theoretic construction is required; any gap here would invalidate the radial SLE identification.

Authors: We agree that a more explicit treatment of the existence and uniqueness of the canonical three-pointed LQG surface is warranted. The manuscript invokes this surface by building directly on the measure-theoretic constructions from the chordal case in Sheffield (2010) and subsequent works on LQG with marked points. In the revision we will add to §2 both a precise citation to those results and a brief self-contained paragraph explaining how the radial three-pointed configuration is obtained from the chordal one via a standard conformal mapping that introduces no new parameters, thereby making the load-bearing step fully traceable. revision: yes
Referee: [§4] §4 (imaginary-geometry description of the interface): the claim that the welded curve is radial SLE via imaginary geometry relies on the welding map preserving the LQG structure; the argument must explicitly check that the radial Loewner driving function obtained after welding matches the known law for radial SLE_κ (with the correct κ determined by the LQG parameter γ).

Authors: The identification proceeds from the fact that the welding map preserves the LQG measure and the imaginary-geometry coupling, which determines the driving function. To address the request for an explicit check, the revised §4 will include a short computation verifying that the radial Loewner driving function extracted after welding coincides with the standard radial SLE_κ law (κ = 16/γ²) that is known from the imaginary-geometry literature. This addition will make the matching fully explicit without altering the existing argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extends independent Sheffield result

full rationale

The paper's central claim is an extension of Sheffield's prior independent work on LQG conformal welding to a radial three-pointed configuration. The abstract explicitly grounds the result in that external foundation rather than any self-citation chain, fitted parameter renamed as prediction, or self-definitional construction. No load-bearing step reduces by the paper's own equations or citations to its inputs; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the established theory of LQG measures, conformal welding, and imaginary geometry as background; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Existence and conformal invariance properties of Liouville quantum gravity surfaces and measures
Invoked implicitly as the setting for the welding construction.
domain assumption Well-definedness of conformal welding for the canonical three-pointed surface
Required for the interface to be a radial SLE.

pith-pipeline@v0.9.0 · 5606 in / 1145 out tokens · 22187 ms · 2026-05-23T08:24:51.084127+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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