Domain Markov property of the Gaussian free field

For an open $U\subset D$, the field decomposes as $h=h_U+h^{D\setminus U}$, where $h_U$ is an independent zero-boundary Gaussian free field on $U$ and $h^{D\setminus U}$ is harmonic on $U$ and agrees with the outside field in the distributional sense.

A random set is local for a Gaussian free field if, conditionally on the set and its associated harmonic function, the remaining field is an independent zero-boundary field in its complement. This extends the deterministic Domain Markov property of the Gaussian free field to appropriate random revealed domains. Arbitrary field-dependent random sets need not have this property.