Random Conformal Welding for Finitely Connected Regions

  • Shi-Yi Lan
  • Wang ZhouEmail author


Given a finitely connected region \(\Omega \) 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 \(\partial U_j\) is constructed using the exponential Gaussian free field. The existence and uniqueness of random conformal welding of \(\Omega \) with \(h_j\) is established by investigating a non-uniformly elliptic Beltrami equation with a random complex dilatation. This generalizes the result of Astala, Jones, Kupiainen and Saksman to multiply connected domains.


Random welding Quasiconformal mapping Gaussian free field SLE 

Mathematics Subject Classification (2010)

30C62 60D05 



The authors would like to thank one referee for his/her helpful comments and suggestions.


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Authors and Affiliations

  1. 1.Guangxi University for NationalitiesNanningChina
  2. 2.National University of SingaporeSingaporeSingapore

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