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The Gaussian free field on a transient weighted graph is the centered Gaussian random field whose covariance kernel is the random walk Green function of a transient weighted graph.

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 $h\in H_0$.

Solved by gpt-5.6-sol high.

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 $\mathrm{cap}(\{0\})$.

Solved by gpt-5.6-sol high.