The continuum GFF φ φ on Ω Ω is a Gaussian random field indexed by H 1 ( Ω Ω ) , i.e., a collection of Gaussian random variables, one for each f ∈ ∈ H 1 ( Ω Ω ) , denoted by ⟨ ⟨ φ φ , f ⟩ ⟩ , such that the covariance structure is C o v [ ⟨ ⟨ φ φ , f ⟩ ⟩ , ⟨ ⟨ φ φ , g ⟩ ⟩ ] = ⟨ ⟨ f , g ⟩ ⟩ for all f , g ∈ ∈ H 1 ( Ω Ω ) .