• DOI:10.5948/UPO9781614440222
• Corpus ID: 119671461

Random walks and electric networks

@article{Doyle1987RandomWA,
  title={Random walks and electric networks},
  author={Peter G. Doyle and James Laurie Snell},
  journal={American Mathematical Monthly},
  year={1987},
  volume={94},
  pages={202}
}

• P. Doyle, J. Snell
• Published 1 February 1987
• Mathematics
• American Mathematical Monthly

Probability theory, like much of mathematics, is indebted to physics as a source of problems and intuition for solving these problems. Unfortunately, the level of abstraction of current mathematics often makes it difficult for anyone but an expert to appreciate this fact. In this work we will look at the interplay of physics and mathematics in terms of an example where the mathematics involved is at the college level. The example is the relation between elementary electric network theory and random walks. Central to the work will be Polya’s beautiful theorem 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. Our goal will be to interpret this theorem as a statement about electric networks, and then to prove the theorem using techniques from classical electrical theory. The techniques referred to go back to Lord Rayleigh, who introduced them in connection with an investigation of musical instruments. The analog of Polya’s theorem in this connection is that wind instruments are possible in our three-dimensional world, but are not possible in Flatland (Abbott [1]). The connection between random walks and electric networks has been recognized for some time (see Kakutani [12], Kemeny, Snell, and