 Solution

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.