Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 209 / 3 / 1

Set $\tilde{\Gamma }(x)=\Gamma (x)-\Gamma (b)$. Then $\tilde{\Gamma }(b)=0$, and subtraction of a constant leaves every edge difference unchanged:The linear maphas determinant $\pm 1$: its inverse is $\gamma (x)=\tilde{\gamma }(x)-\tilde{\gamma }(a)$, because $\gamma (a)=0$. The change of variables formula therefore shows that the new density is proportional toon functions vanishing at $b$. This is precisely the discrete Gaussian free field with Dirichlet boundary condition at $b$.