arxiv: 2604.13845 · v1 · submitted 2026-04-15 · 🧮 math.PR · math-ph· math.CV· math.MP

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Minkowski content construction of the CLE gasket measure

Jason Miller , Yizheng Yuan

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Pith reviewed 2026-05-10 12:41 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.CVmath.MP

keywords CLE gasketMinkowski contentconformal loop ensembleconformally covariant measurepercolation scaling limitkappa in (4,8)resistance metric

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The pith

The canonical conformally covariant measure on the CLE gasket equals the limit of its Minkowski content and covering numbers for kappa in (4,8).

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

The paper shows that the measure on the gasket of a conformal loop ensemble with parameter kappa between 4 and 8 arises directly as the scaling limit of several concrete approximations to the size of the gasket set. These include the Euclidean Minkowski content, the count of intersecting dyadic squares after renormalization, and the minimal number of small balls needed to cover the gasket in both the Euclidean metric and the gasket's own geodesic and resistance metrics. As a consequence the construction matches the measure previously obtained from the scaling limit of critical percolation clusters when kappa equals 6. The work also proves that the resulting measure on any fixed compact set has finite moments of every order, extending what was known before.

Core claim

We show for κ ∈ (4,8) that the canonical conformally covariant measure on the CLE_κ gasket, previously constructed indirectly, can be realized as the limit of the Euclidean Minkowski content and its box-count variants, the properly renormalized number of dyadic squares intersecting the gasket, and the properly renormalized minimal covering numbers of balls of radius δ in both the canonical geodesic and resistance metrics. This identifies the CLE_6 gasket measure with the conformally covariant measure constructed by Garban-Pete-Schramm as a scaling limit of the number of vertices in a macroscopic critical percolation cluster. Along the way the gasket measure of every fixed compact set is show

What carries the argument

The renormalized Minkowski content of the CLE gasket, defined as the limit of the Lebesgue measure of its δ-neighborhood (or equivalent covering counts) after multiplication by the appropriate power of δ.

If this is right

The measure equals the limit of renormalized counts of intersecting dyadic squares.
For κ = 6 the measure coincides with the Garban-Pete-Schramm percolation-cluster measure.
The measure on any compact set has finite moments of all orders.
Equivalent constructions hold when coverings are taken with respect to the gasket's geodesic or resistance metric.

Where Pith is reading between the lines

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

Grid-based counting of squares at fine scales now supplies a practical way to approximate the measure numerically.
Conformal covariance properties already known for Minkowski content transfer directly to the CLE gasket measure.
The identification at κ = 6 supplies a new route to study percolation-cluster scaling limits through the loop-ensemble description.

Load-bearing premise

The earlier indirect construction of the measure by Miller and Schoug is the correct canonical conformally covariant target measure on the gasket.

What would settle it

Compute the limit of the renormalized number of dyadic squares intersecting a fixed compact set inside one fixed CLE realization and verify whether it equals the mass assigned to the same set by the Miller-Schoug indirect construction.

Figures

Figures reproduced from arXiv: 2604.13845 by Jason Miller, Yizheng Yuan.

**Figure 1.1.** Figure 1.1: Illustration of the statement of Proposition 1.5. Shown in red are two simple admissible paths between x and w ∈ ∂Bpath(x, r) in a gasket Υ of Γ. The region Bx,w is a simply connected set in the Euclidean plane and contains the two red paths, hence needs to contain the shaded region between them. Outline. The remainder of this article is structured as follows. In Section 2, we will collect some prelimina… view at source ↗

**Figure 2.1.** Figure 2.1: Illustration of how the chords η can be resampled in D \ K as to link outside K while preserving the gasket ΥI ∩ K (indicated in red). • We have η res ⊆ D \ K. • The components of ΥI ∩ K remain connected to I; the other components of K \ η are inside loops of Γ res. (This is possible since I contains boundary arcs of the same parity.) See [PITH_FULL_IMAGE:figures/full_fig_p010_2_1.png] view at source ↗

read the original abstract

We show for $\kappa \in (4,8)$ that the canonical conformally covariant measure on the conformal loop ensemble (CLE$_\kappa$) gasket, previously constructed indirectly by the first co-author and Schoug, can be realized as the limit of several natural approximation schemes. These include the Euclidean Minkowski content and its box-count variants, the properly renormalized number of dyadic squares that intersect the gasket, and the properly renormalized minimal number of balls of radius $\delta$ necessary to cover the gasket with respect to both its canonical geodesic and resistance metrics. This in particular allows us to identify the CLE$_6$ gasket measure with the conformally covariant measure constructed by Garban-Pete-Schramm as a scaling limit of the number of vertices in a macroscopic critical percolation cluster on the triangular lattice. Along the way, we show that the CLE gasket measure of every fixed compact set has finite moments of all orders; previously this was only known for first moments.

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

Miller and Yuan give explicit Minkowski content and covering approximations that recover the known CLE gasket measure and add all-moment bounds plus a percolation identification.

read the letter

The core advance is that the canonical conformally covariant measure on the CLE_κ gasket for κ in (4,8) can be obtained directly as the limit of Minkowski content, renormalized dyadic square counts, and minimal coverings in the geodesic and resistance metrics. For κ=6 this recovers the Garban-Pete-Schramm percolation measure as well. They also prove that the measure on any fixed compact set has finite moments of all orders, which was previously known only for the first moment. These are concrete, usable constructions rather than another existence proof. The work sits on top of the earlier indirect construction by Miller and Schoug and the standard existence and covariance properties of CLE_κ, so the new technical content is the convergence theorems and the moment estimates. The approximations are natural geometric ones, which makes the result potentially useful for simulation or further analysis in random planar geometry. A minor point worth checking in the proofs is that the renormalization constants really are fixed independently of the target measure; the abstract presents them as coming from the approximation schemes themselves, but any hidden fitting would weaken the claim. No other gaps jump out from the stated results. This paper is aimed at people already working on conformal loop ensembles, percolation scaling limits, or random geometry who need a more hands-on version of the gasket measure. It is not a broad survey, but specialists will find the explicit realizations and the higher-moment result worth having. It deserves a serious referee because the claims are specific, the connections to percolation are clean, and the moment improvement is a clear step forward. I would send it out for review.

Referee Report

0 major / 0 minor

Summary. The paper proves that for κ ∈ (4,8), the canonical conformally covariant measure on the CLE_κ gasket (previously constructed indirectly by Miller and Schoug) arises as the limit of several explicit approximation schemes: the Euclidean Minkowski content and its box-count variants, the renormalized count of intersecting dyadic squares, and the renormalized minimal covering numbers by δ-balls in the canonical geodesic and resistance metrics. It also establishes that the gasket measure on any fixed compact set has finite moments of all orders (extending the known first-moment result) and identifies the κ=6 case with the Garban-Pete-Schramm conformally covariant measure obtained from critical percolation on the triangular lattice.

Significance. If the limit theorems and moment bounds hold, the work supplies direct, constructive realizations of the CLE gasket measure that are independent of the prior indirect definition. This strengthens the analytic toolkit for studying conformally invariant random measures, enables explicit computations, and confirms the equivalence of the Miller-Schoug and Garban-Pete-Schramm constructions at κ=6. The all-order moment result is a concrete advance with potential applications to tail estimates and concentration.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, for the accurate summary of our results, and for the positive recommendation to accept. We are pleased that the referee views the constructive realizations of the CLE gasket measure and the all-order moment bounds as significant contributions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes convergence of explicit approximation schemes (Minkowski content, dyadic counts, covering numbers in geodesic/resistance metrics) to the CLE gasket measure. These limit theorems and moment bounds rest on established CLE_κ existence and covariance properties, plus an external identification for κ=6 with the Garban-Pete-Schramm percolation measure. The prior Miller-Schoug construction is cited only for identification of the target; the new proofs supply independent content and do not reduce to self-definition, fitted inputs, or unverified self-citation chains. No equations or steps in the supplied abstract or context exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence and conformal covariance of CLE_κ for κ in (4,8) and on the correctness of the earlier indirect construction of the gasket measure.

axioms (2)

domain assumption Existence of the conformal loop ensemble CLE_κ with the required conformal covariance for κ ∈ (4,8)
Invoked as the object whose gasket measure is being constructed.
domain assumption The measure previously constructed by Miller and Schoug is the canonical conformally covariant gasket measure
The paper takes this as the target that the new limits must match.

pith-pipeline@v0.9.0 · 5464 in / 1262 out tokens · 41196 ms · 2026-05-10T12:41:53.661455+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 3 canonical work pages · 1 internal anchor

[1]

The Ising magnetisation field and the Gaussian free field

Tom \'a s Alcalde L \'o pez , Lorca Heeney , and Marcin Lis . The Ising magnetisation field and the Gaussian free field . arXiv e-prints , page arXiv:2602.05886, February 2026

work page arXiv 2026
[2]

A continuous proof of the existence of the SLE _8 curve

Valeria Ambrosio and Jason Miller . A continuous proof of the existence of the SLE _8 curve . arXiv e-prints , page arXiv:2203.13805, March 2022

work page arXiv 2022
[3]

Multiple SLE _ from CLE _ for (4,8)

Valeria Ambrosio, Jason Miller, and Yizheng Yuan. Multiple SLE _ from CLE _ for (4,8) . ArXiv e-prints , 2025

2025
[4]

Tightness of approximations to metrics on non-simple conformal loop ensemble gaskets

Valeria Ambrosio, Jason Miller, and Yizheng Yuan. Tightness of approximations to metrics on non-simple conformal loop ensemble gaskets. ArXiv e-prints , 2025

2025
[5]

The scaling limit of critical I sing interfaces is CLE_3

St\' e phane Benoist and Cl\' e ment Hongler. The scaling limit of critical I sing interfaces is CLE_3 . Ann. Probab. , 47(4):2049--2086, 2019

2049
[6]

Conformal measure ensembles for percolation and the FK - I sing model

Federico Camia, Ren\'e Conijn, and Demeter Kiss. Conformal measure ensembles for percolation and the FK - I sing model. In Sojourns in probability theory and statistical physics. II . B rownian web and percolation, a F estschrift for C harles M . N ewman , volume 299 of Springer Proc. Math. Stat. , pages 44--89. Springer, Singapore, [2019] 2019

2019
[7]

Federico Camia and Charles M. Newman. Two-dimensional critical percolation: the full scaling limit. Comm. Math. Phys. , 268(1):1--38, 2006

2006
[8]

The scaling limit of random walk and the intrinsic metric on planar critical percolation

Irina D ankovi\' c , Maarten Markering, Jason Miller, and Yizheng Yuan. The scaling limit of random walk and the intrinsic metric on planar critical percolation. 2026

2026
[9]

Pivotal, cluster, and interface measures for critical planar percolation

Christophe Garban, G\'abor Pete, and Oded Schramm. Pivotal, cluster, and interface measures for critical planar percolation. J. Amer. Math. Soc. , 26(4):939--1024, 2013

2013
[10]

The M inkowski content measure for the L iouville quantum gravity metric

Ewain Gwynne and Jinwoo Sung. The M inkowski content measure for the L iouville quantum gravity metric. Ann. Probab. , 52(2):658--712, 2024

2024
[11]

Analysis on fractals , volume 143 of Cambridge Tracts in Mathematics

Jun Kigami. Analysis on fractals , volume 143 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2001

2001
[12]

Conformal removability of non-simple Schramm-Loewner evolutions

Konstantinos Kavvadias , Jason Miller , and Lukas Schoug . Conformal removability of non-simple Schramm-Loewner evolutions . arXiv e-prints , page arXiv:2302.10857, February 2023

work page internal anchor Pith review Pith/arXiv arXiv 2023
[13]

Conformal invariance of boundary touching loops of FK I sing model

Antti Kemppainen and Stanislav Smirnov. Conformal invariance of boundary touching loops of FK I sing model. Comm. Math. Phys. , 369(1):49--98, 2019

2019
[14]

Lawler and Mohammad A

Gregory F. Lawler and Mohammad A. Rezaei. Minkowski content and natural parameterization for the S chramm- L oewner evolution. Ann. Probab. , 43(3):1082--1120, 2015

2015
[15]

Lawler and Scott Sheffield

Gregory F. Lawler and Scott Sheffield. A natural parametrization for the S chramm- L oewner evolution. Ann. Probab. , 39(5):1896--1937, 2011

1937
[16]

Conformal restriction: the chordal case

Gregory Lawler, Oded Schramm, and Wendelin Werner. Conformal restriction: the chordal case. J. Amer. Math. Soc. , 16(4):917--955, 2003

2003
[17]

Lawler, Oded Schramm, and Wendelin Werner

Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. , 32(1B):939--995, 2004

2004
[18]

Lawler and Wang Zhou

Gregory F. Lawler and Wang Zhou. SLE curves and natural parametrization. Ann. Probab. , 41(3A):1556--1584, 2013

2013
[19]

Imaginary geometry I : interacting SLE s

Jason Miller and Scott Sheffield. Imaginary geometry I : interacting SLE s. Probab. Theory Related Fields , 164(3-4):553--705, 2016

2016
[20]

Imaginary geometry III : reversibility of SLE_ for (4,8)

Jason Miller and Scott Sheffield. Imaginary geometry III : reversibility of SLE_ for (4,8) . Ann. of Math. (2) , 184(2):455--486, 2016

2016
[21]

Gaussian free field light cones and SLE _ ( )

Jason Miller and Scott Sheffield. Gaussian free field light cones and SLE _ ( ) . Ann. Probab. , 47(6):3606--3648, 2019

2019
[22]

Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles

Jason Miller and Lukas Schoug. Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles. Ann. Inst. Henri Poincar\' e Probab. Stat. , 60(4):2267--2296, 2024

2024
[23]

Jason Miller, Nike Sun, and David B. Wilson. The H ausdorff dimension of the CLE gasket. Ann. Probab. , 42(4):1644--1665, 2014

2014
[24]

C LE percolations

Jason Miller, Scott Sheffield, and Wendelin Werner. C LE percolations. Forum Math. Pi , 5:e4, 102, 2017

2017
[25]

Non-simple SLE curves are not determined by their range

Jason Miller, Scott Sheffield, and Wendelin Werner. Non-simple SLE curves are not determined by their range. J. Eur. Math. Soc. (JEMS) , 22(3):669--716, 2020

2020
[26]

Non-simple conformal loop ensembles on L iouville quantum gravity and the law of CLE percolation interfaces

Jason Miller, Scott Sheffield, and Wendelin Werner. Non-simple conformal loop ensembles on L iouville quantum gravity and the law of CLE percolation interfaces. Probab. Theory Related Fields , 181(1-3):669--710, 2021

2021
[27]

Simple conformal loop ensembles on L iouville quantum gravity

Jason Miller, Scott Sheffield, and Wendelin Werner. Simple conformal loop ensembles on L iouville quantum gravity. Ann. Probab. , 50(3):905--949, 2022

2022
[28]

Connection probabilities for conformal loop ensembles

Jason Miller and Wendelin Werner. Connection probabilities for conformal loop ensembles. Comm. Math. Phys. , 362(2):415--453, 2018

2018
[29]

Existence and uniqueness of the canonical B rownian motion in non-simple conformal loop ensemble gaskets

Jason Miller and Yizheng Yuan. Existence and uniqueness of the canonical B rownian motion in non-simple conformal loop ensemble gaskets. ArXiv e-prints , 2025

2025
[30]

Existence and uniqueness of the conformally covariant geodesic metric on non-simple conformal loop ensemble gaskets

Jason Miller and Yizheng Yuan. Existence and uniqueness of the conformally covariant geodesic metric on non-simple conformal loop ensemble gaskets. ArXiv e-prints , 2025

2025
[31]

Basic properties of SLE

Steffen Rohde and Oded Schramm. Basic properties of SLE . Ann. of Math. (2) , 161(2):883--924, 2005

2005
[32]

Scaling limits of loop-erased random walks and uniform spanning trees

Oded Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. , 118:221--288, 2000

2000
[33]

Exploration trees and conformal loop ensembles

Scott Sheffield. Exploration trees and conformal loop ensembles. Duke Math. J. , 147(1):79--129, 2009

2009
[34]

Critical percolation in the plane: conformal invariance, C ardy's formula, scaling limits

Stanislav Smirnov. Critical percolation in the plane: conformal invariance, C ardy's formula, scaling limits. C. R. Acad. Sci. Paris S\' e r. I Math. , 333(3):239--244, 2001

2001
[35]

Oded Schramm, Scott Sheffield, and David B. Wilson. Conformal radii for conformal loop ensembles. Comm. Math. Phys. , 288(1):43--53, 2009

2009
[36]

Oded Schramm and David B. Wilson. S LE coordinate changes. New York J. Math. , 11:659--669, 2005

2005
[37]

Conformal loop ensembles: the M arkovian characterization and the loop-soup construction

Scott Sheffield and Wendelin Werner. Conformal loop ensembles: the M arkovian characterization and the loop-soup construction. Ann. of Math. (2) , 176(3):1827--1917, 2012

1917