Chordal restriction property

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