On the equivalence of cylinder tilings and planar electric networks

In the famous paper (Brooks et al., 1940) an equivalence was established between planar electric networks and tilings of cylinders. However, the proof given there is rather difficult and requires knowledge of a larger theory. We give a simple new proof of this equivalence. Furthermore, there is a well known relationship between electric networks and random walks. We show how the tiling associated with the network corresponding to a random walk can shed light on various aspects of the walk, such as hitting times.