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A Book: Random walks and electric networks
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Random walks and electric networks by Peter G. Doyle, J. Laurie Snell Hardcover: 159 pages Publisher: Mathematical Assn of America (December 1984) Language: English ISBN: 0883850249 Product Dimensions: 0.8 x 5.2 x 7.5 inches The book brings together two of my passions : random walks and electric networks. It turns out that there are interesting relationships between these two areas, so insights in one provide can be used to prove things in the other. There is this beautiful theorem by Polya which states that a random walker on an infinite street network in d-dimensional space is bound to return to the starting point when d = 2, but has a positive probability of escaping to infinity without returning to the starting point when d >= 3. The book reinterprets this theorem as a statement about electric networks, and then proves the theorem using techniques from classical network theory. The proof relies on showing that the resistance of the corresponding electric network in 1 and 2 dimensions is infinite, whereas it is finite in the 3 dimensional case. Thus some current [like our random walker] can flow to infinity http://www.ee.technion.ac.il/~adam/FUN/RWEN.pdf |
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