Conformal invariance of SLE

Chordal SLE is defined on any simply connected domain with two marked boundary points by applying a conformal map from the upper half-plane. Its law is independent of the chosen map, up to a deterministic change of parameterization.

A scale is a conformal bijection sending a marked starting boundary point to zero and target boundary point to infinity in the complex upper half-plane. A curve is SLE in that scale when its image is ordinary SLE using the half-plane-capacity parameterization of that image.

A random curve $\gamma$ from $0$ to infinity in the complex upper half-plane has the chordal restriction property when, for every admissible compact H-hull $A$ avoiding zero, the conditional law of ${\psi }_A(\gamma )$ given $\gamma \cap A=\varnothing$ equals the original law of $\gamma$. Here ${\psi }_A:\mathbb{H}\setminus A\to \mathbb{H}$ fixes zero and has derivative one at infinity.

A conformally invariant random chordal filling has exponent $\alpha$ when its avoidance probability for every admissible hull $A$ is the displayed power, where ${\Phi }_A$ fixes zero and has derivative one at infinity. Composition of two hull-removing maps and the chain rule make conditional image avoidance probabilities the same as unconditional ones. The avoidance probabilities determine a chordal filling law principle therefore implies the chordal restriction property. Independent unions followed by filling add exponents, since avoidance probabilities multiply.

A chordal filling is the closed connected set obtained from a proper boundary-to-boundary path by adding complementary pockets disconnected from both boundary arcs between its marked endpoints. In $(\mathbb{H},0,\infty )$ it meets the boundary only at zero and infinity and leaves two complementary simply connected sides, adjacent to the negative and positive real axes. Equivalently it is full with respect to the two marked boundary arcs. This definition uses the prime ends of a general simply connected domain and is preserved by conformal maps.

The probabilities that a random chordal filling avoids each admissible compact H-hull away from zero determine its law as an unparameterized closed set. Each point of its complement can be joined to the relevant real boundary arc through the complement and lies in a thin rational polygonal tube attached to that arc. These countably many hull tests recover the complement. Their joint avoidance events are tested by filled finite unions; if a union separates zero from infinity, the joint avoidance event is empty. Otherwise connectedness makes avoidance of the union equivalent to avoidance of its filled hull. Thus individual hull probabilities give every finite joint indicator distribution and determine the random complement.

Suppose a random proper simple curve from zero to infinity has the displayed avoidance probability for every admissible compact H-hull $A$, with fixed $\alpha >0$. Put ${\psi }_A=g_A-g_A(0)$. For another admissible hull $B$ in the image complex upper half-plane, map composition and the chain rule give $g_C^{\prime }(0)=g_B^{\prime }(0)g_A^{\prime }(0)$ for $C=A\cup {\psi }_A^{-1}(B)$. The conditional image curve therefore has avoidance probabilities $g_B^{\prime }(0{)}^{\alpha }$, exactly those of the original. Since avoidance probabilities determine a simple chordal curve law, the conditional image law agrees with the original. This is the chordal restriction property.

For random proper simple curves from $0$ to infinity in the complex upper half-plane, the probabilities of avoiding every compact H-hull whose closure excludes zero determine the unparameterized trace law. Every point off a simple chordal curve can be connected to a real boundary interval away from zero within one side of its complement. A countable collection of thin rational polygonal tubes attached to that interval therefore recovers the complement from its avoidance indicators. Joint avoidance of finitely many tubes reduces to avoidance of their filled union. If the union blocks every path from zero to infinity the joint event is empty; otherwise filling pockets does not change the avoidance event. The inclusion-exclusion principle gives all finite indicator distributions. These determine the random complement and hence the trace. Simplicity and properness are hypotheses; the assertion is not a general determination theorem for arbitrary random closed sets.

For chordal ${\mathrm{SLE}}_{8/3}$ and an admissible hull $A$, let ${\psi }_t$ map out the image of $A$ after removing the SLE hull. Thenstopped when the trace meets $A$ is a bounded martingale. Its terminal value yields the avoidance probability

The locality property says that an SLE does not detect a hull lying away from its starting point until the curve reaches that hull. Among chordal Schramm-Loewner evolutions, this property holds for ${\mathrm{SLE}}_6$.

A scale-invariant law of unparameterized chords from zero to infinity in the complex upper half-plane is local when a boundary-preserving conformal map between neighborhoods of zero, fixing zero, transports its curve stopped on leaving the first neighborhood to the same-law curve stopped on leaving the second. Equality concerns the stopped curves up to increasing reparameterization. For SLE with parameter six, the transformed SLE driving function is a continuous local martingale with bracket six times the image capacity clock. The Lévy characterization of Brownian motion makes that driver Brownian in the new clock, proving the local stopped-curve invariance.

Changing the target boundary point by a conformal automorphism of the upper half-plane preserves the local chordal SLE law at parameter six, up to a capacity time change and until the relevant boundary point is swallowed. The transformed SLE driving function has zero drift at six; the Dambis-Dubins-Schwarz theorem makes it Brownian in image capacity time.

At parameter six, changing a boundary target preserves the initial unparameterized SLE law until the growing hull separates the two possible targets. After separation, each curve continues in the complementary component whose boundary contains its own target. A curve targeted at infinity follows the unbounded component, whereas a finite separated target lies on a different component. The target-change locality of SLE6 proof no longer applies: its conformal automorphism of the upper half-plane conjugates corresponding surviving domains only before this separation. A zero drift before contact does not give equality of complete curve laws afterwards.