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PREFACE

Peter G. Doyle
Affiliation:
Dartmouth College
J. Laurie Snell
Affiliation:
Dartmouth College
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Summary

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 book 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 book will be Pólya'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 Pólya'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 [13], Kemeny, Snell, and Knapp [15], and Kelly [14]).

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Publisher: Mathematical Association of America
Print publication year: 1984

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  • PREFACE
  • Peter G. Doyle, Dartmouth College, J. Laurie Snell, Dartmouth College
  • Book: Random Walks and Electric Networks
  • Online publication: 05 January 2012
  • Chapter DOI: https://doi.org/10.5948/UPO9781614440222.001
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  • PREFACE
  • Peter G. Doyle, Dartmouth College, J. Laurie Snell, Dartmouth College
  • Book: Random Walks and Electric Networks
  • Online publication: 05 January 2012
  • Chapter DOI: https://doi.org/10.5948/UPO9781614440222.001
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • PREFACE
  • Peter G. Doyle, Dartmouth College, J. Laurie Snell, Dartmouth College
  • Book: Random Walks and Electric Networks
  • Online publication: 05 January 2012
  • Chapter DOI: https://doi.org/10.5948/UPO9781614440222.001
Available formats
×