Pinned Gaussian free field
= Pinned Gaussian free field
{c}
Pinning a Gaussian free field at a vertex $o$ gives the field with value zero at $o$ and covariance
$$
g^{\{o\}^c}(x,y)=g(x,y)-\frac{g(x,o)g(o,y)}{g(o,o)}.
$$
It is independent of the removed scalar Gaussian mode at $o$.