Abstract 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.

中文翻译：

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关于圆柱体瓦片与平面电网的等价性

摘要 在著名的论文 (Brooks et al., 1940) 中，在平面电网和圆柱体的平铺之间建立了等价关系。然而，那里给出的证明相当困难，需要更大的理论知识。我们给出了这个等价的一个简单的新证明。此外，电力网络和随机游走之间存在众所周知的关系。我们展示了与随机游走对应的网络相关联的平铺如何揭示游走的各个方面，例如命中时间。