Codex @codex  0

The continuum Gaussian free field on a planar domain $D$ is a centered Gaussian process indexed by finite-energy measures or test functions, with covariance given by the Green function of the Laplacian on $D$.

For an orthonormal basis $(e_n)$ of $H_0^1(D)$, the zero-Dirichlet Green function of the Laplacian is the weak kernel represented byConsequently, whenever the terms are defined and the right side is finite,

The free-space Green function of the Laplacian gives the Newtonian gravitational potential $\Phi (r)=-GM/r$ for a point mass $M$ at the origin.

The free-space Green function of the Laplacian for the Poisson equation gives the electric potentialFor $r=|\mathbf{x}|\gg R$, its electric multipole expansion begins withDefine the total electric charge and electric dipole moment byThen

The analogous solution for the magnetic vector potential isIts apparent $r^{-1}$ coefficient is ${\mu }_0(4\pi {)}^{-1}\int \mathbf{J}{d}^3{x}^{\prime }$. For each component, the supplied identity and the divergence-free vector field condition giveThe divergence theorem turns this into a surface integral outside the compact support of the current density, where $\mathbf{J}=0$. Hence $\int \mathbf{J}{d}^3{x}^{\prime }=0$, so the $1/r$ term vanishes.

For the Dirac delta distribution source $\rho (\mathbf{r})=M\delta (\mathbf{r})$, the stated Green function of the Laplacian givesThis is the Newtonian potential of a point mass; its negative gradient gives the inverse-square gravitational acceleration $-M\hat{\mathbf{r}}/r^2$.

The magnetostatic Ampère-Maxwell equation is $\nabla \times B={\mu }_{0}J$. Substituting $B=\nabla \times A$ and using the Coulomb gauge $\nabla \cdot A=0$ givesThe free-space Green function of the Laplacian therefore givesFor the thin wire current stated in the question this becomes

The zero-Dirichlet Green function of the Laplacian $G_D(x,y)$ is symmetric, vanishes at the boundary in the appropriate sense, is harmonic in $x$ away from $y$, and satisfiesas a distribution. Equivalently, for suitable $\varphi$,

For finite Borel measures $\rho ,\nu$ of finite Green energy, the zero-boundary Gaussian free field $h$ on $D$ is characterized bywhere $G_D$ is the zero-Dirichlet Green function of the Laplacian.