Existence and Uniqueness of Nonsimple Multiple SLE

We prove the existence and uniqueness of multiple \(\hbox {SLE}_{\kappa }\) ${\text{SLE}}_{\kappa }$ associated with any given link pattern for \(\kappa \in (4,6]\) $\kappa \in (4,6]$ . We also have the uniqueness for \(\kappa \in (6,8)\) $\kappa \in (6,8)$ . The multiple \(\hbox {SLE}_{\kappa }\) ${\text{SLE}}_{\kappa }$ law is constructed by first inductively constructing a \(\sigma \) $\sigma$ -finite multiple \(\hbox {SLE}_{\kappa }\) ${\text{SLE}}_{\kappa }$ measure and then normalizing the measure whenever it is finite. The total mass of the measure satisfies the conformal covariance, asymptotics and PDE for multiple \(\hbox {SLE}_{\kappa }\) ${\text{SLE}}_{\kappa }$ partition functions in the literature subject to the assumption that it is smooth.