Tobias Weihrauch, Stefan Bachmann
Published 2019-02-04Version 1
On a finite weighted graph $G$, the effective resistance between two vertices $x$ and $y$ admits the probabilistic representation \[ R(x,y) = \frac{1}{c_x \cdot \mathbb{P}_x[\tau_y < \tau_x^+]} \] using the the random walk $\mathbb{P}_x$ on $G$ starting in $x$. We show that the same representation does not hold in general for the free effective resistance of an infinite graph. More precisely, we show that a transient graph admits such a representation if and only if it is a subgraph of an infinite line.
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