= Directed edge occupation in a random-walk commute
{title2=$\mathbb E S(u,v)=R_{\mathrm{eff}}(a,z)$}
Let <simple random walk> travel from $a\ne z$ to its first visit to $z$, then stop on its subsequent first hit of $a$ in a finite connected unweighted loopless <graph>. Every directed <edge> $u\to v$ has expected traversal count
$$
\mathbb E S(u,v)=R_{\mathrm{eff}}(a,z).
$$
The <Strong Markov property> splits the count into the two killed legs. By <killed-walk occupation voltage>, their contributions are $v_{az}(u)$ and $v_{za}(u)$. Their Laplacians cancel, so the <harmonic maximum principle on a finite graph> makes the sum constant, with value $v_{az}(a)+v_{za}(a)=R_{\mathrm{eff}}(a,z)$. Summing over all directed <edges> recovers the <commute time identity>.
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