Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 209 / 3 / 2 / Solution

Only the $2d$ edges incident to $x_0$ depend on $\Gamma (x_0)$. Conditional on the remaining field, writeCompleting the square givesThe conditional density is therefore proportional to $e^{-(u-m{)}^2}$, and henceThis is the Gibbs-Markov property of the discrete Gaussian free field in the normalization used by the paper.