 Solution

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.