Random conformal welding for finitely connected regions

Given a finitely connected region $\mathrm{\Omega }$ of the Riemann sphere whose complement consists of $m$ mutually disjoint closed disks ${\bar{U}}_j$, the random homeomorphism $h_j$ on the boundary component $\mathrm{\partial }{U}_j$ is constructed using the exponential Gaussian free field. The existence and uniqueness of random conformal welding of $\mathrm{\Omega }$ with $h_j$ is established by investigating a non-uniformly elliptic Betrami equation with a random complex dilatation. This generalizes the result of Astala, Jones, Kupiainen and Saksman to multiply connected domains.