Extending work of Foster, Doyle, and others, we show how the Foster theorems, a family of results concerning effective resistances on finite graphs, can in certain cases be extended to infinite graphs. A family of sum rules is then obtained, which allows one to easily calculate the sum of the resistances over all paths of a given length. The results are illustrated with some of the most common grids in the plane, including the square, triangular, and hexagonal grids.
|Number of pages||14|
|Journal||Journal of Statistical Mechanics: Theory and Experiment|
|Publication status||Published - 3 Apr 2017|
- stochastic processes