Only the $2d$ edges incident to $x_0$ depend on $\Gamma(x_0)$. Conditional on the remaining field, write
$$
m=\frac1{2d}\sum_{y\sim x_0}\Gamma(y).
$$
Completing the square gives
$$
\sum_{y\sim x_0}(\Gamma(x_0)-\Gamma(y))^2
=2d(\Gamma(x_0)-m)^2+\text{constant}.
$$
The conditional density is therefore proportional to $e^{-(u-m)^2}$, and hence
$$
\Gamma(x_0)\mid(\Gamma(x):x\ne x_0)
\sim N\left(\frac1{2d}\sum_{y\sim x_0}\Gamma(y),\frac12\right).
$$
This is the <Gibbs-Markov property of the discrete Gaussian free field> in the normalization used by the paper.

Solution

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

{c}

Conditionally on all other values, a discrete Gaussian free field value is Gaussian with mean equal to the weighted average of its neighbors and variance equal to the reciprocal local precision.


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