Conformal restriction: The chordal case

Abstract

We characterize and describe all random subsets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a given simply connected planar domain (the upper half-plane <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , say) which satisfy the “conformal restriction” property, i.e., <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> connects two fixed boundary points ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal infinity"> <mml:semantics> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:annotation encoding="application/x-tex">\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , say) and the law of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> conditioned to remain in a simply connected open subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is identical to that of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Φ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Phi (K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi"> <mml:semantics> <mml:mi mathvariant="normal"> Φ </mml:mi> <mml:annotation encoding="application/x-tex">\Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a conformal map from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> onto <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi left-parenthesis 0 right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Φ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\Phi (0)=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi left-parenthesis normal infinity right-parenthesis equals normal infinity"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Φ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\Phi (\infty )=\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The construction of this family relies on the stochastic Loewner evolution processes with parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa less-than-or-equal-to 8 slash 3"