Schramm-Loewner evolution contains a topological Sierpiński carpet when $κ$ is close to 8

TL;DR

This work establishes that SLE$_\kappa$ curves with $\kappa$ near 8 almost surely have range structures containing a topological Sierpiński carpet, achieved by coupling SLE with the Gaussian free field via imaginary geometry and embedding Mandelbrot-type fractal percolation into the SLE trace. The main tool is a multi-scale fractal-percolation argument adapted to the dependent, IG-informed SLE setting, supported by precise regularity estimates as $\kappa\to 8$ and a coarse-grained analysis of the GFF. Consequently, the trace is almost surely conformally non-removable and the conformal welding problem for such SLEs has a non-unique solution; moreover, the adjacency graph of bubbles becomes disconnected in this regime. These results link fractal-percolation phenomena to the fine topological structure of SLE traces and have implications for the conformal welding and the broader understanding of SLE in the near-critical regime.

Abstract

We consider the Schramm-Loewner evolution (SLE$_κ$) for $κ\in (4,8)$, which is the regime where the curve is self-intersecting but not space-filling. We show that there exists $δ_0>0$ such that for $κ\in (8 - δ_0,8)$, the range of an SLE$_κ$ curve almost surely contains a topological Sierpiński carpet. Combined with a result of Ntalampekos (2021), this implies that in this parameter range, SLE$_κ$ is almost surely conformally non-removable, and the conformal welding problem for SLE$_κ$ does not have a unique solution. Our result also implies that for $κ\in (8 - δ_0,8)$, the adjacency graph of the complementary connected components of the SLE$_κ$ curve is disconnected.

Figures (10)

• Figure 1: SLE$_\kappa$ curves arise as conformal welding interfaces Sheffield-zipperDMS21-LQG-MRT. Left: The case $\kappa \in (0,4]$. Right: The case $\kappa \in (4,8)$.
• Figure 2: $N=4$ and on the event that $[0,1]^2$ is $\infty$-good. The regions in light-blue (resp. in light-blue or orange) are $[0,1]^2 \setminus A_1^\dagger$ (resp. $[0,1]^2 \setminus A_2^\dagger$). The set $A_2^*$ is obtained from $A_1^* \cap A_2^\dagger$ by further removing the green boxes.
• Figure 3: The boundaries of two removed box clusters intersect at some point. Decreasing $p$ by any positive amount will almost surely merge these two clusters into one.
• Figure 4: Left: The region in gray (resp. in gray or light-gray) represents the range of $\widetilde{\eta}$ stopped upon hitting $u$ (resp. $v$). In this figure, the point $u$ lies on the trace of $\eta$ whereas $v$ does not. Right: In Definition \ref{['def:n-good']}, the 0-good property requires that flow lines with different angles, after leaving the $\epsilon$-neighborhoods of starting points (marked in bold), are mutually non-intersecting. In this figure, the pair $(u_1,v_1)$ is admissible whereas $(u_2,v_2)$ is not.
• Figure 5: Left: The exceptional point $v \in \eta^L(u)$ in Property \ref{['property:ig-dense']}. After hitting $v$ (and having filled the gray region), the curve $\widetilde{\eta}$ proceeds to fill $D_1$ (in light-blue) and then $D_2$ (in pink). Right: An illustration for the proof of Lemma \ref{['lem:flow-line-not-touch']}. If $\eta^L(w)$ and $\eta^R(w)$ intersect at $v \neq w$, then we can construct $w_1,w_2 \in B(x^{(n)},3\epsilon^n)$ that violates the $n$-good property of $x^{(n)}$.

Theorems & Definitions (66)

• Theorem 1.1: Ntalampekos-carpet
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Conjecture 1.7
• Proposition 2.1
• proof : Proof of Proposition \ref{['prop:carpet-fractal']}
• Proposition 2.2