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We solve the classical conformal welding problem for a composition of tworandom homeomorphisms generated by independent Gaussian multiplicative chaosmeasures with small parameter values. In other words, given two such measureson the boundary of the unit disk we show that there exist conformal maps tocomplementary domains on the Riemann sphere such that the pushforward of thenormalised measures agree on their common boundary.

Conformal welding of independent Gaussian multiplicative chaos measures