Random Walks and Electrical Networks (Carus Mathematical Monographs)

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.

Strongly recommended!. Also check out Peter Doyle's webpage
at Dartmouth "http://math.dartmouth.edu/~doyle/"


Relationship between Random Walks and Electric Networks!   January 27, 2002
 4 out of 4 found this review helpful

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 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.

Strongly recommended!.


cool analogies   April 1, 2001
This book provides fascinating insights and analogies between random walks and electric networks- and how you can exploit these analogies to solve difficult problems in one using the other... there's also a nice proof of the "Polya's theorem" using these analogies- basically Polya's theorem says that a random walk in dimensions >2 is transient, while a random walk on a plane or a line always returns to its starting point...