Gaussian free field (GFF). As surveyed in [@sheffield2007gaussian], the GFF is a random tempered distribution on ${\mathbb{R}}^d$ (defined modulo additive constant if $d=2$) with covariance given by $$\operatorname{Cov}[(h,\phi_1),(h,\phi_2)] =
\int_{{\mathbb{R}}^d}\int_{{\mathbb{R}}^d}\Phi(x-y)\phi_1(x)\phi_2(y)\,dx\,dy,$$ where $\Phi$ is the fundamental solution of the Laplace equation in ${\mathbb{R}}^d$. The two-dimensional GFF (which is the same as $\operatorname{FGF}_1({\mathbb{R}}^2)$) has been studied in a wide range of contexts in recent years. It can be obtained as a scaling limit of random discrete models, such as domino tilings [@kenyon2001dominos], as well as continuum models, such as those arising in random matrix theory [@rider2006noise]. It is central to conformal field theory and Liouville quantum gravity [@sheffield2010conformal; @duplantier2011liouville] and has many connections to the Schramm Loewner evolution [@dubedat2009sle; @schramm2010contour; @millerimaginary1; @millerimaginary2; @millerimaginary3; @millerimaginary4]. The 2D GFF is also known in the geostatistics literature as the *de Wijs process* or the *logarithmic variogram model*, where it was introduced in the early 1950s to describe ore deposits [@de1951statistics; @de1953statistics; @mondalapplying; @chiles2009geostatistics]. More recently, variations in crop yields have been modeled using the GFF [@mccullagh2002statistical; @mccullagh2006evidence].

For all $d\in {\mathbb{N}}$, the $d$-dimensional GFF exhibits a certain Markov property: For each fixed domain $D \subset {\mathbb{R}}^d$, if we are given the restriction a GFF $h$ 