Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 5 / A2

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.

On the segment the energy containswhich diverges logarithmically along the diagonal. The uniform segment measure therefore has infinite Green energy, so its mean cannot be defined as an $L^2$ Gaussian-field pairing.