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arxiv: 2302.10857 · v2 · submitted 2023-02-21 · 🧮 math.PR · math-ph· math.CV· math.MP

Conformal removability of non-simple Schramm-Loewner evolutions

Konstantinos Kavvadias , Jason Miller , Lukas Schoug This is my paper

Pith reviewed 2026-05-24 09:36 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.CVmath.MP
keywords SLESchramm-Loewner evolutionconformal removabilitycut pointsrandom curvesplanar fractalscomplementary components
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The pith

The range of an SLE_κ curve is almost surely conformally removable for κ in the set K where the adjacency graph of complementary components is connected.

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

The paper proves that for κ between 4 and 8 where the adjacency graph of the connected components of the complement of an SLE_κ curve is almost surely connected, the range of the curve is conformally removable almost surely. This directly answers a question posed by Sheffield. The argument proceeds by first constructing a canonical conformally covariant volume measure supported on the cut points of the curve for all κ in (4,8) and proving that this measure obeys an upper bound controlled by the diameter of any Borel set. Gwynne and Pfeffer had already established that the set K of such κ is non-empty. The connectedness of the adjacency graph then allows the measure to be used to establish removability of the entire range.

Core claim

For every κ belonging to the set K, the range of an SLE_κ curve is almost surely conformally removable. The set K consists of those κ in (4,8) for which the adjacency graph on the connected components of the complement is almost surely connected, meaning any two complementary components can be joined by a finite chain of components whose boundaries intersect. The proof constructs the canonical conformally covariant volume measure on the cut points and shows it satisfies a precise diameter bound, after which graph connectedness implies the removability statement.

What carries the argument

The adjacency graph of connected components of the complement of the SLE_κ curve, whose almost-sure connectedness, combined with the diameter-bounded conformally covariant measure on cut points, implies that the range is conformally removable.

If this is right

  • The range of SLE_κ is conformally removable almost surely for every κ in the non-empty set K.
  • A canonical conformally covariant volume measure on cut points exists for all κ in (4,8).
  • This measure assigns to any Borel set a mass bounded above by a function of the set's diameter.
  • Connectivity of the adjacency graph is sufficient to promote the cut-point measure into a proof of removability of the full range.

Where Pith is reading between the lines

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

  • If further work shows that K equals the entire interval (4,8), then every non-simple non-space-filling SLE curve would have a conformally removable range.
  • The same graph-connectivity criterion may apply to other planar random curves whose complementary components admit an adjacency graph.
  • The diameter bound on the cut-point measure supplies a quantitative tool that could be used to study other geometric properties such as dimension or intersection exponents.

Load-bearing premise

The canonical conformally covariant volume measure on the cut points of SLE_κ exists for κ in (4,8) and satisfies the required upper bound in terms of diameter.

What would settle it

An explicit construction showing that the candidate measure on cut points fails to be conformally covariant or violates the diameter upper bound for some κ in (4,8), or an example of a connected adjacency graph whose corresponding SLE range fails to be removable.

Figures

Figures reproduced from arXiv: 2302.10857 by Jason Miller, Konstantinos Kavvadias, Lukas Schoug.

Figure 1
Figure 1. Figure 1: Illustration of the setup of the proof of Lemma 3.7. The map φm takes the component of Czpη1pr0, τ1,msq Y η2pr0, τ2,msqq containing w to H as shown. The flow lines on the left side are generated using the field h and the flow lines on the right side are generated using the field rh on H, which we compare to hm “ h˝φ ´1 m ´χ argpφ ´1 m q 1 in the proof. Proof of Lemma 3.7. Step 1. Setup. Fix δ P p0, 1q smal… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the setup of Lemma 4.1. We note that Lemma 4.1 does not assume that X is connected and in fact when we use Lemma 4.1 to prove Theorem 1.1 we will be in the situation that X is totally disconnected as shown [PITH_FULL_IMAGE:figures/full_fig_p037_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the setup of Lemma 4.2, which bounds the number of squares in a Whitney square decomposition of D of side length 2´j which are hit by the collection of hyperbolic geodesics in D from z to a point in X (such geodesics are shown in red and the squares of side length 2´j which are hit are in blue) in terms of the upper Minkowski dimension of X. H for some w P Xu. Let also ϕ : D Ñ D be the conf… view at source ↗
Figure 4
Figure 4. Figure 4: Left: Illustration of the set Xa. Shown in red (resp. blue) are the flow lines with angle ´π{2 (resp. π{2) starting from the grid of points A2 X paZq 2 , a ą 0 small, stopped upon exiting A2. These paths together make up Xa. Right: Illustration of the construction of Ya from Xa. Shown in light green is a component of A2zXa whose boundary consists of four arcs, given by the left/right sides of flow lines of… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the statement of Lemma 5.1. Shown in red (resp. blue) is A1 (resp. A2). The points zj and paths γj are shown but unlabeled in the figure. Lemma 5.1. Suppose that κ 1 P K. Fix p P p0, 1q and b P p0, dcut κ1 q. There exists a0 P p0, 1q and C0 ą 0 such that whenever a P p0, a0q and C ě C0 the following holds with probability at least p. There exist connected components U1, . . . , Um of A2zYa … view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of the setup of the proof of Lemma 5.11. Shown in blue is a chordal SLEκ1pκ 1 ´ 6q process in D from ´i to i up to some time t. The point Xt is the most recent intersection of η 1 |r0,ts with the counterclockwise arc of BD from ´i to i. Shown in red is the part of η 1 starting from time t up until τt , which is the first time that it disconnects Xt from i. The law of η 1 |rt,τts is equal to th… view at source ↗
read the original abstract

We consider the Schramm-Loewner evolution (SLE$_\kappa$) for $\kappa \in (4,8)$, which is the regime that the curve is self-intersecting but not space-filling. We let ${\mathcal K}$ be the set of $\kappa \in (4,8)$ for which the adjacency graph of connected components of the complement of an SLE$_\kappa$ is a.s. connected, meaning that for every pair of complementary components $U, V$ there exist complementary components $U_1,\ldots,U_n$ with $U_1 = U$, $U_n = V$, and $\partial U_i \cap \partial U_{i+1} \neq \emptyset$ for each $1 \leq i \leq n-1$. It was proved by Gwynne and Pfeffer that this set is non-empty. We show that the range of an SLE$_\kappa$ for $\kappa \in {\mathcal K}$ is a.s. conformally removable, which answers a question of Sheffield. As a step in the proof, we construct the canonical conformally covariant volume measure on the cut points of an SLE$_\kappa$ for $\kappa \in (4,8)$ and establish a precise upper bound on the measure that it assigns to any Borel set in terms of its diameter.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that for κ belonging to the non-empty set K ⊂ (4,8) on which the adjacency graph of complementary components of an SLE_κ curve is a.s. connected (a fact due to Gwynne-Pfeffer), the range of the curve is a.s. conformally removable. The argument proceeds by constructing a canonical conformally covariant volume measure supported on the cut points of SLE_κ (κ ∈ (4,8)) and establishing an upper bound on the measure of any Borel set in terms of its diameter; connectivity of the adjacency graph together with this measure is then used to deduce removability.

Significance. If the central claims hold, the result answers an open question of Sheffield on conformal removability for non-simple SLEs and supplies a new conformally covariant measure on cut points that may be of independent interest in the study of SLE geometry. The combination of the Gwynne-Pfeffer connectivity result with the new measure construction yields a clean implication from graph-theoretic properties to analytic removability.

major comments (1)
  1. [measure construction section] The construction of the canonical conformally covariant measure on cut points (final paragraph of the abstract and the corresponding section containing its definition) is load-bearing: the removability implication requires both a.s. support on cut points and the stated diameter upper bound to control the relevant Hausdorff content. The derivation of the bound and the verification that the measure is supported on the cut-point set (including handling of null sets) must be checked for completeness and uniformity of constants.
minor comments (2)
  1. [Introduction] Notation for the set K and the adjacency graph should be introduced with a numbered definition or displayed equation for easy reference in later sections.
  2. [main theorem statement] The statement of the main removability theorem would benefit from an explicit citation to the precise removability criterion (e.g., the Hausdorff-content condition) being applied.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. We address the single major comment below.

read point-by-point responses

  1. Referee: [measure construction section] The construction of the canonical conformally covariant measure on cut points (final paragraph of the abstract and the corresponding section containing its definition) is load-bearing: the removability implication requires both a.s. support on cut points and the stated diameter upper bound to control the relevant Hausdorff content. The derivation of the bound and the verification that the measure is supported on the cut-point set (including handling of null sets) must be checked for completeness and uniformity of constants.

    Authors: We agree that the measure construction is load-bearing for the removability argument. The canonical conformally covariant measure on cut points is constructed in Section 3 via a weak-limit procedure from approximations along cut times of the SLE curve; conformal covariance is obtained in Proposition 3.3 by direct use of the conformal invariance of the driving Brownian motion. Almost-sure support on the cut-point set is established in Proposition 3.6, which shows that the measure of the (random) set of non-cut points is zero by combining the definition with the fact that non-cut points form a null set for the underlying SLE measure; null sets are handled uniformly via the almost-sure statements that hold simultaneously for all rational domains by countable union. The diameter upper bound appears as Theorem 3.9 and is derived from the Hölder continuity of the SLE curve together with the Markov property, yielding a constant that depends only on κ and is uniform across domains. While we believe these arguments are complete as written, we will add a short clarifying subsection (3.4) that explicitly records the uniformity of constants and the null-set handling to facilitate verification. revision: partial

Circularity Check

0 steps flagged

No significant circularity; external citation and novel construction

full rationale

The paper cites Gwynne-Pfeffer (distinct authors) only for non-emptiness of K and constructs a new conformally covariant measure on cut points with an explicit diameter bound as an independent intermediate step. The removability claim for κ in K is derived from graph connectivity plus this measure's properties; no equation, definition, or self-citation reduces the target result to its own inputs by construction. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on standard domain assumptions of SLE theory for κ ∈ (4,8) and on the prior result that K is non-empty; the new measure is constructed rather than postulated.

axioms (1)
  • domain assumption Standard properties of SLE_κ for κ ∈ (4,8): the curve is self-intersecting but not space-filling and its complement components are well-defined.
    Invoked throughout the abstract to set the regime of the result.
invented entities (1)
  • Canonical conformally covariant volume measure on cut points no independent evidence
    purpose: To obtain a precise upper bound on the measure of any Borel set in terms of its diameter
    Constructed in the paper as an intermediate step; no independent evidence outside the construction is supplied in the abstract.

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Forward citations

Cited by 2 Pith papers

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    The CLE gasket measure for κ in (4,8) equals the Minkowski content limit and related covering approximations.

  2. Existence and uniqueness of the canonical Brownian motion in non-simple conformal loop ensemble gaskets

    math.PR 2025-12 unverdicted novelty 8.0

    Existence and uniqueness of a canonical Brownian motion on CLE_κ gaskets for κ ∈ (4,8), characterized by a unique locally determined resistance form satisfying invariance properties.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages · cited by 2 Pith papers · 1 internal anchor

  1. [1]

    Ambrosio and J

    V. Ambrosio and J. Miller. A continuous proof of the existence of the SLE 8 curve. arXiv e-prints , page arXiv:2203.13805, Mar. 2022

    work page arXiv 2022

  2. [2]

    V. Beffara. The dimension of the SLE curves. Ann. Probab., 36(4):1421–1452, 2008

    work page 2008

  3. [4]

    S. Benoist. Natural parametrization of SLE: the Gaussian free field point of view. Electron. J. Probab., 23:Paper No. 103, 16, 2018

    work page 2018

  4. [5]

    J. Bertoin. L´ evy processes, volume 121. Cambridge university press Cambridge, 1996

    work page 1996

  5. [6]

    Duplantier, J

    B. Duplantier, J. Miller, and S. Sheffield. Liouville quantum gravity as a mating of trees.Ast´ erisque, (427):viii+257, 2021

    work page 2021

  6. [7]

    Duplantier and S

    B. Duplantier and S. Sheffield. Liouville quantum gravity and KPZ. Invent. Math., 185(2):333–393, 2011

    work page 2011

  7. [8]

    G. B. Folland. Real analysis. Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, second edition, 1999. Modern techniques and their applications, A Wiley-Interscience Publication

    work page 1999

  8. [9]

    J. B. Garnett and D. E. Marshall. Harmonic measure, volume 2 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2008. Reprint of the 2005 original

    work page 2008

  9. [10]

    Gwynne, N

    E. Gwynne, N. Holden, and J. Miller. An almost sure KPZ relation for SLE and Brownian motion. Ann. Probab., 48(2):527–573, 2020

    work page 2020

  10. [11]

    Gwynne and J

    E. Gwynne and J. Miller. Convergence of the self-avoiding walk on random quadrangulations to SLE 8{3 ona 8{3-Liouville quantum gravity. Ann. Sci. ´Ec. Norm. Sup´ er. (4), 54(2):305–405, 2021

    work page 2021

  11. [12]

    Gwynne and J

    E. Gwynne and J. Miller. Percolation on uniform quadrangulations and SLE 6 on a 8{3-Liouville quantum gravity. Ast´ erisque, (429):vii+242, 2021

    work page 2021

  12. [13]

    Gwynne, J

    E. Gwynne, J. Miller, and W. Qian. Conformal invariance of CLE κ on the riemann sphere for κ P p4, 8q. International Mathematics Research Notices, 2021(23):17971–18036, 2021

    work page 2021

  13. [14]

    Gwynne, J

    E. Gwynne, J. Miller, and X. Sun. Almost sure multifractal spectrum of Schramm-Loewner evolution. Duke Math. J., 167(6):1099–1237, 2018

    work page 2018

  14. [15]

    Gwynne and J

    E. Gwynne and J. Pfeffer. Connectivity properties of the adjacency graph of SLEκ bubbles for κPp 4, 8q. Ann. Probab., 48(3):1495–1519, 2020

    work page 2020

  15. [16]

    Holden and E

    N. Holden and E. Powell. Conformal welding for critical Liouville quantum gravity. Ann. Inst. Henri Poincar´ e Probab. Stat., 57(3):1229–1254, 2021

    work page 2021

  16. [17]

    P. W. Jones and S. K. Smirnov. Removability theorems for Sobolev functions and quasiconformal maps. Ark. Mat., 38(2):263–279, 2000

    work page 2000

  17. [18]

    Kallenberg

    O. Kallenberg. Random measures, theory and applications , volume 77 of Probability Theory and Stochastic Modelling. Springer, Cham, 2017

    work page 2017

  18. [19]

    Kallenberg

    O. Kallenberg. Foundations of modern probability , volume 99 of Probability Theory and Stochastic Modelling . Springer, Cham, [2021] ©2021. Third edition [of 1464694]

    work page 2021

  19. [20]

    Regularity of the SLE$_4$ uniformizing map and the SLE$_8$ trace

    K. Kavvadias, J. Miller, and L. Schoug. Regularity of the SLE4 uniformizing map and the SLE8 trace. arXiv e-prints, page arXiv:2107.03365, July 2021. 70 KONSTANTINOS KAVVADIAS, JASON MILLER, LUKAS SCHOUG

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  20. [21]

    Kavvadias, J

    K. Kavvadias, J. Miller, and L. Schoug. Conformal removability of SLE 4. arXiv e-prints, page arXiv:2209.10532, Sept. 2022

    work page arXiv 2022

  21. [22]

    Kenyon, J

    R. Kenyon, J. Miller, S. Sheffield, and D. B. Wilson. Bipolar orientations on planar maps and SLE12. Ann. Probab., 47(3):1240–1269, 2019

    work page 2019

  22. [23]

    Koskela and T

    P. Koskela and T. Nieminen. Quasiconformal removability and the quasihyperbolic metric. Indiana Univ. Math. J., 54(1):143–151, 2005

    work page 2005

  23. [24]

    Lawler, O

    G. Lawler, O. Schramm, and W. Werner. Conformal restriction: the chordal case. J. Amer. Math. Soc. , 16(4):917– 955, 2003

    work page 2003

  24. [25]

    G. F. Lawler. Conformally invariant processes in the plane . Number 114. American Mathematical Soc., 2008

    work page 2008

  25. [26]

    G. F. Lawler and M. A. Rezaei. Minkowski content and natural parameterization for the Schramm-Loewner evolution. Ann. Probab., 43(3):1082–1120, 2015

    work page 2015

  26. [27]

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

    work page 2004

  27. [28]

    G. F. Lawler and S. Sheffield. A natural parametrization for the Schramm-Loewner evolution. Ann. Probab., 39(5):1896–1937, 2011

    work page 1937

  28. [29]

    G. F. Lawler and W. Zhou. SLE curves and natural parametrization. Ann. Probab., 41(3A):1556–1584, 2013

    work page 2013

  29. [30]

    Y. Li, X. Sun, and S. S. Watson. Schnyder woods, SLE(16), and Liouville quantum gravity. arXiv e-prints, page arXiv:1705.03573, May 2017

    work page arXiv 2017

  30. [31]

    Miller and W

    J. Miller and W. Qian. The geodesics in Liouville quantum gravity are not Schramm-Loewner evolutions. Probab. Theory Related Fields, 177(3-4):677–709, 2020

    work page 2020

  31. [32]

    Miller and L

    J. Miller and L. Schoug. Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles. arXiv e-prints, page arXiv:2201.01748, Jan. 2022

    work page arXiv 2022

  32. [33]

    Miller and S

    J. Miller and S. Sheffield. Imaginary geometry I: interacting SLEs.Probab. Theory Related Fields, 164(3-4):553–705, 2016

    work page 2016

  33. [34]

    Miller and S

    J. Miller and S. Sheffield. Imaginary geometry III: reversibility of SLE κ for κ P p4, 8q. Ann. of Math. (2) , 184(2):455–486, 2016

    work page 2016

  34. [35]

    Miller and S

    J. Miller and S. Sheffield. Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees. Probab. Theory Related Fields, 169(3-4):729–869, 2017

    work page 2017

  35. [36]

    Miller and S

    J. Miller and S. Sheffield. Gaussian free field light cones and SLE κpρq. Ann. Probab., 47(6):3606–3648, 2019

    work page 2019

  36. [37]

    Miller, S

    J. Miller, S. Sheffield, and W. Werner. CLE percolations. Forum Math. Pi, 5:e4, 102, 2017

    work page 2017

  37. [38]

    Miller, N

    J. Miller, N. Sun, and D. B. Wilson. The Hausdorff dimension of the CLE gasket. Ann. Probab., 42(4):1644–1665, 2014

    work page 2014

  38. [39]

    Miller and H

    J. Miller and H. Wu. Intersections of SLE paths: the double and cut point dimension of SLE. Probab. Theory Related Fields, 167(1-2):45–105, 2017

    work page 2017

  39. [40]

    Nacu and W

    c. Nacu and W. Werner. Random soups, carpets and fractal dimensions. J. Lond. Math. Soc. (2) , 83(3):789–809, 2011

    work page 2011

  40. [41]

    Ntalampekos

    D. Ntalampekos. Non-removability of the Sierpi´ nski gasket.Invent. Math., 216(2):519–595, 2019

    work page 2019

  41. [42]

    Ntalampekos

    D. Ntalampekos. Non-removability of Sierpi´ nski carpets.Indiana Univ. Math. J. , 70(3):847–854, 2021

    work page 2021

  42. [43]

    M. A. Rezaei and D. Zhan. Higher moments of the natural parameterization for SLE curves. Ann. Inst. Henri Poincar´ e Probab. Stat., 53(1):182–199, 2017

    work page 2017

  43. [44]

    Robert and V

    R. Robert and V. Vargas. Gaussian multiplicative chaos revisited. The Annals of Probability , 38(2):605–631, 2010

    work page 2010

  44. [45]

    Rohde and O

    S. Rohde and O. Schramm. Basic properties of SLE. Ann. of Math. (2) , 161(2):883–924, 2005

    work page 2005

  45. [46]

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

    work page 2000

  46. [47]

    Schramm and S

    O. Schramm and S. Sheffield. Contour lines of the two-dimensional discrete Gaussian free field. Acta Math., 202(1):21–137, 2009

    work page 2009

  47. [48]

    Schramm, S

    O. Schramm, S. Sheffield, and D. B. Wilson. Conformal radii for conformal loop ensembles. Comm. Math. Phys. , 288(1):43–53, 2009

    work page 2009

  48. [49]

    Schramm and D

    O. Schramm and D. B. Wilson. SLE coordinate changes. New York J. Math. , 11:659–669, 2005

    work page 2005

  49. [50]

    S. Sheffield. Gaussian free fields for mathematicians. Probab. Theory Related Fields, 139(3-4):521–541, 2007

    work page 2007

  50. [51]

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

    work page 2009

  51. [52]

    S. Sheffield. Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Ann. Probab., 44(5):3474–3545, 2016

    work page 2016

  52. [53]

    S. Sheffield. Quantum gravity and inventory accumulation. Ann. Probab., 44(6):3804–3848, 2016. CONFORMAL REMOVABILITY OF NON-SIMPLE SCHRAMM-LOEWNER EVOLUTIONS 71

    work page 2016

  53. [54]

    Sheffield and W

    S. Sheffield and W. Werner. Conformal loop ensembles: the Markovian characterization and the loop-soup construction. Ann. of Math. (2) , 176(3):1827–1917, 2012

    work page 1917

  54. [55]

    S. Smirnov. Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris S´ er. I Math., 333(3):239–244, 2001

    work page 2001

  55. [56]

    S. Smirnov. Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2), 172(2):1435–1467, 2010

    work page 2010