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If a measure $\rho$ has finite Green energy, thenconverges in $L^2$ for any orthonormal basis $(e_n)$ of $H_0^1(D)$ and independent standard normal coefficients ${\xi }_n$. The limit is centered Gaussian with variance equal to the Green energy of $\rho$ and does not depend on the chosen basis.

Choose an orthonormal basis $(e_n)$ of $H_0^1(D)$ and write the GFF formally as $h={\sum }_n{\xi }_n{e}_n$, where the ${\xi }_n$ are independent standard normal variables. The Green-kernel expansion in the Dirichlet space givesConsequently the seriesconverges in $L^2$. Its partial sums are centered Gaussian and their variances converge to the displayed Green energy. The $L^2$ limit is therefore Gaussian with mean zero and varianceThis constructs the Finite-Green-energy measure pairing with a Gaussian free field independently of the chosen orthonormal basis.

The Zero-boundary Gaussian free field on the unit disc is the centered Gaussian process indexed by finite Borel measures of finite Green energy, with covarianceThis covariance determines all its finite-dimensional distributions.

The Domain Markov property of the Gaussian free field says that for every suitable open $U\subset \mathbb{D}$ one can writewhere $h_U$ is a zero-boundary Gaussian free field on $U$, independent of $h^{\mathbb{D}\setminus U}$, while $h^{\mathbb{D}\setminus U}$ is harmonic on $U$ and carries the information from the field outside $U$.

A finite measure $\mu$ can index the field when its Green energyis finite. On the two-dimensional square, the singularity is locally integrable because polar area contributes $rdr$ while $G$ contributes $1/r$. The near-diagonal integral is therefore proportional to ${\int }_0^{\epsilon }dr<\infty$, and the mean $\Gamma (\mu )$ is a well-defined centered Gaussian random variable.