Chordal SLE$_6$ explorations of a quantum disk

Abstract

We consider a particular type of $\sqrt{8/3} $-Liouville quantum gravity surface called a doubly marked quantum disk (equivalently, a Brownian disk) decorated by an independent chordal SLE$_6$ curve $\eta $ between its marked boundary points. We obtain descriptions of the law of the quantum surfaces parameterized by the complementary connected components of $\eta ([0,t])$ for each time $t \geq 0$ as well as the law of the left/right $\sqrt{8/3} $-quantum boundary length process for $\eta $.