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Random Conformal Welding for Finitely Connected Regions

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Abstract

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.

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Acknowledgements

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

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Authors and Affiliations

  1. Guangxi University for Nationalities, Nanning, China

    Shi-Yi Lan

  2. National University of Singapore, Singapore, Singapore

    Wang Zhou

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  1. Shi-Yi Lan
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  2. Wang Zhou
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Corresponding author

Correspondence to Wang Zhou.

Additional information

Shi-Yi Lan was partially supported by the NSF of China (11661011) and NSF of Guangxi (2016GXNSFAA380099). Wang Zhou was partially supported by a Grant R-155-000-192-114 at the National University of Singapore.

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Lan, SY., Zhou, W. Random Conformal Welding for Finitely Connected Regions. J Theor Probab 32, 659–683 (2019). https://doi.org/10.1007/s10959-018-0874-5

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  • DOI: https://doi.org/10.1007/s10959-018-0874-5

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