Conformal welding problem, flow line problem, and multiple Schramm--Loewner evolution

Abstract

A quantum surface (QS) is an equivalence class of pairs $(D,H)$ of simply connected domains $D\subsetneq\mathbb{C}$ and random distributions $H$ on $D$ induced by the conformal equivalence for random metric spaces. This distribution-valued random field is extended to a QS with $N+1$ marked boundary points (MBPs) with $N\in\mathbb{Z}_{\ge 0}$. We propose the conformal welding problem for it in the case of $N\in\mathbb{Z}_{\ge 1}$. If $N=1$, it is reduced to the problem introduced by Sheffield, who solved it by coupling the QS with the Schramm--Loewner evolution (SLE). When $N \ge 3$, there naturally appears room of making the configuration of MBPs random, and hence a new problem arises how to determine the probability law of the configuration. We report that the multiple SLE in $\mathbb{H}$ driven by the Dyson model on $\mathbb{R}$ helps us to fix the problems and makes them solvable for any $N \ge 3$. We also propose the flow line problem for an imaginary surface with boundary condition changing points (BCCPs). In the case when the number of BCCPs is two, this problem was solved by Miller and Sheffield. We address the general case with an arbitrary number of BCCPs in a similar manner to the conformal welding problem. We again find that the multiple SLE driven by the Dyson model plays a key role to solve the flow line problem.