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[交流] A Book: Random walks and electric networks

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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