Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 226 / 2

The Dirichlet energy space with zero boundary at infinity iswhere $C_c(G)$ is the vector space of finitely supported real functions on the vertices. Transience makes the Dirichlet form of a Markov chain positive definite on this completion.

The Strong Markov property at the first visit to $0$ gives the singleton equilibrium potential from the Green functionLet $U_n$ be an increasing exhaustion of $G$ by finite sets containing $0$, and let $g_{U_n}(x,0)$ be the Green function of a transient weighted graph stopped on leaving $U_n$. Each $g_{U_n}(\cdot ,0)$ has finite support. The Green identity and the Markov property give, for $m\ge n$,By the monotone convergence theorem, the right side tends to zero as $m,n\to \infty$, while $g_{U_n}(x,0)\uparrow g(x,0)$. Thus $g(\cdot ,0)$ is an $H_0$ limit of finitely supported functions. Dividing by the positive number $g(0,0)$ proves $h\in H_0$.

For the normalized Green function of a transient weighted graph, the Green identity isTaking $f=g(\cdot ,0)$ and using $h=g(\cdot ,0)/g(0,0)$ therefore givesThis is also the capacity of a finite set for a transient random walk $\mathrm{cap}(\{0\})$.

The field $\psi$ is centered and jointly Gaussian. Since the Gaussian free field has covariance $\mathbb{E}[{\phi }_x{\phi }_y]=g(x,y)$ and $h_x=g(x,0)/g(0,0)$,Also ${\psi }_0=0$ andJoint Gaussianity turns this zero covariance into independence. Therefore $\psi$ is a Pinned Gaussian free field at $0$, independent of ${\phi }_0$, with covariance equal to the Green function of the walk killed on hitting $0$.