A two-valued local set of a planar Gaussian free field is a random closed set whose conditional
harmonic function has only two prescribed values
in its complementary components. For the zero-boundary field with covariance 
, the canonical thin set with values
and 
is defined for 
and 
. After subtracting the respective harmonic
value, the remaining fields in the complementary components are independent zero-boundary
Gaussian free fields. The set includes the
domain boundary and is a measurable function
of the field.
Two-valued local sets can be explored successively inside complementary components using the zero-boundary Gaussian free fields supplied by the spatial Markov property. Such iterations occur in representations of critical Ising model magnetisation fields (Alcalde López et al. 2026).
