Gaussian free field

Remarks

The exponent equals $-\frac12 \langle h, -\Delta_U h \rangle$, and $-\Delta_U$ is positive definite because $B$ is nonempty and $G$ is connected. So $h$ restricted to $U$ is a centred Gaussian process with covariance matrix $(-\Delta_U)^{-1}$. This matrix is the Green function of the continuous-time random walk that crosses each edge at rate $1$ and is killed on $B$. Example: let $V = \set{a, x, b}$ with edges $\set{a, x}$ and $\set{x, b}$, and $B = \set{a, b}$. The exponent is $-\frac12 (h(x)^2 + h(x)^2) = -h(x)^2$, so $h(x)$ is normal with mean $0$ and variance $1/2$, and indeed $-\Delta_U = (2)$ has inverse $1/2$. Non-example: with $B$ empty, the exponent is unchanged when a constant is added to $h$, so no normalising constant $Z$ exists. The continuum Gaussian free field on a domain $D \subseteq \R^d$ is the centred Gaussian random generalised function whose covariance is the Dirichlet Green function of $-\Delta$ on $D$. It is not a function when $d \ge 2$.