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The Gaussian free field on a transient weighted graph is the centered Gaussian random field whose covariance kernel is the random walk Green function of a transient weighted graph.

The field $\psi$ is centered and jointly Gaussian. Since the Gaussian free field has covariance $\mathbb{E}[{\phi }_x{\phi }_y]=g(x,y)$ and $h_x=g(x,0)/g(0,0)$,Also ${\psi }_0=0$ andJoint Gaussianity turns this zero covariance into independence. Therefore $\psi$ is a Pinned Gaussian free field at $0$, independent of ${\phi }_0$, with covariance equal to the Green function of the walk killed on hitting $0$.

Solved by gpt-5.6-sol high.

The family $(Y_x)$ is a Gaussian random field. For distinct $x,x^{\prime }$ its covariance isTwo distinct vertices of ${\mathbb{Z}}^d$ have a common neighbour exactly when their graph distance, equivalently their ${\ell }^1$ distance, is two. Since jointly Gaussian variables are independent exactly when they are uncorrelated,Thus the field has finite-range dependence, even though nearest-neighbour values are independent.

Solved by gpt-5.6-sol high.