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arxiv: 2107.03365 · v2 · submitted 2021-07-07 · 🧮 math.PR · math-ph· math.CV· math.MP

Regularity of the SLE₄ uniformizing map and the SLE₈ trace

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

Pith reviewed 2026-05-24 13:46 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.CVmath.MP
keywords SLE4SLE8modulus of continuityuniformizing maptraceconformal removabilityJones-Smirnov conditionLoewner evolution
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The pith

The SLE4 uniformizing map has modulus of continuity (log δ^{-1})^{-1/3+o(1)} as δ → 0.

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

The paper determines the precise modulus of continuity for the SLE4 uniformizing map, showing it behaves like (log δ^{-1})^{-1/3+o(1)} as δ approaches zero. This leads directly to the conclusion that the Jones-Smirnov condition for conformal removability does not hold for SLE4. The analysis also establishes that the modulus of continuity for the SLE8 trace under capacity time parameterization is (log δ^{-1})^{-1/4+o(1)}, confirming a prior conjecture.

Core claim

The central claim is that the modulus of continuity of the SLE4 uniformizing map is given by (log δ^{-1})^{-1/3+o(1)} as δ → 0. As a consequence of the analysis, the Jones-Smirnov condition for conformal removability with quasihyperbolic geodesics does not hold for SLE4. The modulus of continuity for SLE8 with the capacity time parameterization is given by (log δ^{-1})^{-1/4+o(1)} as δ → 0.

What carries the argument

Loewner evolution driven by Brownian motion at parameters 4 and 8, used to construct the uniformizing map and trace and to derive their modulus of continuity estimates.

If this is right

  • The Jones-Smirnov condition for conformal removability does not hold for SLE4.
  • The conjecture of Alvisio and Lawler on the SLE8 trace modulus of continuity is proved.
  • These statements supply sharp quantitative regularity for the SLE4 uniformizing map and the SLE8 trace.

Where Pith is reading between the lines

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

  • The same style of modulus estimates may extend to other specific values of the SLE parameter.
  • Failure of the Jones-Smirnov condition for SLE4 may influence removability questions for related random sets.
  • The results supply a benchmark for numerical checks of Loewner-driven curves.

Load-bearing premise

The analysis relies on the standard construction of SLE via Loewner evolution driven by Brownian motion at the specific parameters 4 and 8, together with the applicability of the Jones-Smirnov condition to the resulting random sets.

What would settle it

A direct computation or high-precision simulation of the Loewner evolution that produces a modulus of continuity with an exponent other than -1/3+o(1) for the SLE4 uniformizing map would contradict the stated result.

Figures

Figures reproduced from arXiv: 2107.03365 by Jason Miller, Konstantinos Kavvadias, Lukas Schoug.

Figure 1
Figure 1. Figure 1: Schematic illustration of how the sections of the paper fit together. 1.3. Outline and proof strategy. The remainder of this article is structured as follows. We will collect a number of preliminaries in Section 2. In Section 3 we establish a version of one of the welding results from [9] in the critical case that κ = 4 and in Section 4 we prove bounds for exit times for SLE when it is parameterized by qua… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the setup to prove Theorems 1.1 and 1.2. We begin with the case κ = 4. Suppose that η is an SLE4 in D from −i to i and DL is the component of D \ η which is to the left of η. Fix  > 0. Suppose that η passes through a point z and w ∈ DL has distance of order  to z. Let φ be the conformal transformation DL → D which fixes −i, −1, and i. Then • the derivative of φ at w is up to constants giv… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the setup to prove Theorem 1.3. We now describe the strategy in the case κ = 8. Suppose that η 0 is an SLE8 in H from 0 to ∞ and let (gt) be its associated Loewner flow. Let also ft = gt − Wt be the centered Loewner flow. Fix t ∈ [0, 1], ζ > 1, and  > 0. It follows from [13] that if we let τ = inf{s ≥ t : |η 0 (s) − η 0 (t)| = } then the event that there exists w with B(w, ζ ) ⊆ η 0 ([t… view at source ↗
read the original abstract

We show that the modulus of continuity of the SLE$_4$ uniformizing map is given by $(\log \delta^{-1})^{-1/3+o(1)}$ as $\delta \to 0$. As a consequence of our analysis, we show that the Jones-Smirnov condition for conformal removability (with quasihyperbolic geodesics) does not hold for SLE$_4$. We also show that the modulus of continuity for SLE$_8$ with the capacity time parameterization is given by $(\log \delta^{-1})^{-1/4+o(1)}$ as $\delta \to 0$, proving a conjecture of Alvisio and Lawler.

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

0 major / 2 minor

Summary. The paper proves that the modulus of continuity of the SLE₄ uniformizing map is (log δ^{-1})^{-1/3+o(1)} as δ→0. As a consequence, the Jones-Smirnov condition for conformal removability fails for SLE₄. It also establishes that the modulus of continuity of the SLE₈ trace under capacity parameterization is (log δ^{-1})^{-1/4+o(1)} as δ→0, confirming a conjecture of Alvisio and Lawler. The arguments rely on the standard Loewner evolution driven by Brownian motion at κ=4 and κ=8 together with the Jones-Smirnov condition applied to the resulting random sets.

Significance. If the results hold, they supply sharp logarithmic moduli of continuity for SLE at the critical parameters κ=4 and κ=8. The failure of the Jones-Smirnov condition for SLE₄ has direct implications for questions of conformal removability, while the SLE₈ result resolves a stated conjecture. The proofs rest on the canonical Loewner-Brownian construction without introducing new free parameters or ad-hoc entities.

minor comments (2)
  1. [Introduction] The introduction would benefit from a short roadmap indicating which sections contain the key estimates controlling the o(1) terms in the moduli statements.
  2. Clarify the precise definition of the uniformizing map and the capacity-time parameterization early in the text to avoid any ambiguity when the moduli are stated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the results, and recommendation of minor revision. The report lists no major comments, so we have no specific points requiring response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained on standard SLE inputs

full rationale

The paper derives moduli of continuity for the SLE₄ uniformizing map and SLE₈ trace directly from the standard Loewner-Brownian construction at fixed κ=4,8 together with the Jones-Smirnov condition applied to the resulting random sets. No quoted step reduces a claimed prediction to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness theorem, or renames an input as an output. The central claims are presented as consequences of analysis on externally defined objects, making the derivation self-contained against the usual SLE benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is available from the abstract alone.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Conformal removability of non-simple Schramm-Loewner evolutions

    math.PR 2023-02 unverdicted novelty 8.0

    For κ in K (where the adjacency graph of SLE_κ complementary components is a.s. connected), the range of SLE_κ is a.s. conformally removable; a conformally covariant measure on cut points is constructed as an intermed...

Reference graph

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