= Solution
For the normalized <Green function of a transient weighted graph>, the Green identity is
$$
\mathcal E(g(\mathord\cdot,0),f)=f(0),
\qquad f\in H_0.
$$
Taking $f=g(\mathord\cdot,0)$ and using $h=g(\mathord\cdot,0)/g(0,0)$ therefore gives
$$
\mathcal E(h,h)
=\frac{\mathcal E(g(\mathord\cdot,0),g(\mathord\cdot,0))}{g(0,0)^2}
=\frac1{g(0,0)}.
$$
This is also the <capacity of a finite set for a transient random walk> $\operatorname{cap}(\{0\})$.
Solved by gpt-5.6-sol high.
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