[Paper Review] Duality of Chordal SLE, II

Dapeng Zhan|ArXiv.org|Mar 14, 2008

Stochastic processes and statistical mechanics8 references4 citations

TL;DR

This paper establishes the duality of chordal SLE for $κ \in (4,8)$ by showing that the boundary of a standard chordal SLE($\kappa$) hull at the random time when a fixed $x \in \mathbb{R} \setminus \{0\}$ is swallowed is distributed as a random integral of SLE($16/\kappa; \vec{\rho}$) traces started from $x$. The key result extends Duplantier's duality conjecture to non-space-filling SLE, using conditioning on the limit of the SLE trace and advanced geometric analysis of SLE($\kappa; \vec{\rho}$) processes.

ABSTRACT

We improve the geometric properties of SLE$(κ;\vecρ)$ processes derived in an earlier paper, which are then used to obtain more results about the duality of SLE. We find that for $κ\in (4,8)$, the boundary of a standard chordal SLE$(κ)$ hull stopped on swallowing a fixed $x\in\R\sem\{0\}$ is the image of some SLE$(16/κ;\vecρ)$ trace started from a random point. Using this fact together with a similar proposition in the case that $κ\ge 8$, we obtain a description of the boundary of a standard chordal SLE$(κ)$ hull for $κ>4$, at a finite stopping time. Finally, we prove that for $κ>4$, in many cases, the limit of a chordal or strip SLE$(κ;\vecρ)$ trace exists.

Motivation & Objective

• To extend Duplantier's duality conjecture for Schramm's SLE to the non-space-filling regime $\kappa \in (4,8)$, where the trace does not visit the swallowed point.
• To resolve the difficulty that for $\kappa \in (4,8)$, the fixed point $x$ is an interior point of the hull boundary, so the boundary cannot be a trace starting from $x$.
• To develop a conditioning mechanism on the limit of the SLE trace at the swallowing time, using integration over the law of $\gamma(T_x)$, to describe the hull boundary as a random mixture of SLE($16/\kappa; \vec{\rho}$) processes.
• To generalize geometric properties of SLE($\kappa; \vec{\rho}$) processes and apply them to prove duality results for $\kappa > 4$.
• To establish the existence of the limit of chordal or strip SLE($\kappa; \vec{\rho}$) traces at finite stopping times for $\kappa > 4$.

Proposed method

• Condition the SLE($\kappa$) process up to time $T_x$ on the value of $\gamma(T_x)$, which is almost surely not a point mass, by expressing the law of the hull as an integral over the law of $\gamma(T_x)$.
• Use the improved geometric properties of SLE($\kappa; \vec{\rho}$) processes derived in Section 4 to analyze the boundary structure of the hull at stopping times.
• Apply a coupling technique from previous work to relate SLE($\kappa_1; \vec{\rho}_1$) and SLE($\kappa_2; \vec{\rho}_2$) with $\kappa_1\kappa_2 = 16$, extending the duality to $\kappa \in (4,8)$.
• Use the conformal invariance and the Loewner equation to relate the boundary of the hull to a SLE($16/\kappa; \vec{\rho}$) trace via a random starting point.
• Analyze the limit behavior of SLE traces in strip and chordal settings using local martingales and conformal maps, particularly $W(z) = 1/\overline{z}$, to study asymptotic behavior at infinity.
• Prove that for $\kappa > 4$, the limit of the SLE($\kappa; \vec{\rho}$) trace exists almost surely at finite stopping times by analyzing the behavior of the driving function and the associated local martingale $h(X(t))$.

Experimental results

Research questions

• RQ1 Can the duality of chordal SLE be extended to the regime $\kappa \in (4,8)$, where the trace is not space-filling and does not visit the swallowed point?
• RQ2 How can the boundary of a chordal SLE($\kappa$) hull at the time when a fixed $x \in \mathbb{R} \setminus \{0\}$ is swallowed be described when the trace does not reach $x$?
• RQ3 What is the correct conditioning mechanism for SLE($\kappa$) processes at the swallowing time of a point, given that the trace almost surely avoids the point?
• RQ4 Does the limit of a chordal or strip SLE($\kappa; \vec{\rho}$) trace exist almost surely at finite stopping times for $\kappa > 4$?
• RQ5 How do the geometric properties of SLE($\kappa; \vec{\rho}$) processes underpin the duality relation between SLE($\kappa$) and SLE($16/\kappa$) in the non-space-filling regime?

Key findings

• For $\kappa \in (4,8)$, the boundary of a standard chordal SLE($\kappa$) hull at the time $T_x$ when a fixed $x \in \mathbb{R} \setminus \{0\}$ is swallowed is distributed as a random integral over $y \in \mathbb{R}$ of SLE($16/\kappa; -\frac{\kappa'}{2}, \frac{3}{2}\kappa' - 4, -\frac{\kappa'}{2} + 2, \kappa' - 4$) traces started from $(y; 0, y^a, y^b, x)$, where $\kappa' = 16/\kappa$, $a = \operatorname{sign}(x)$, $b = \operatorname{sign}(-x)$.
• Almost surely, $\partial K(T_x) \cap \mathbb{H}$ is a crosscut in $\mathbb{H}$ connecting two points $y, z \in \mathbb{R} \setminus \{0\}$ with $\operatorname{sign}(y) = \operatorname{sign}(x)$, $|y| > |x|$, and $\operatorname{sign}(z) = \operatorname{sign}(-x)$.
• For $\kappa \geq 8$, the boundary at swallowing time is almost surely a single SLE($16/\kappa; \vec{\rho}$) trace started from $x$, as shown in Proposition 1.1.
• For $\kappa \in (4,8)$, the hull boundary is not a trace from $x$, but a random mixture of such traces from different starting points, reflecting the fact that $x$ is an interior point of the hull boundary.
• The limit of a chordal or strip SLE($\kappa; \vec{\rho}$) trace exists almost surely at finite stopping times for $\kappa > 4$, as established in Theorem 7.5 and Corollary 7.4.
• In all cases for $\kappa > 4$, the limit of the SLE trace at infinity or at stopping time is almost surely well-defined, with the limit point lying in $(-\infty, p_0)$, $(p_0, \infty)$, or at $\pm\infty$, depending on the force parameters.