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Probability on Graphs: Random Processes on Graphs and Lattices

Geoffrey Grimmett
Cambridge University Press
Publication Date: 
Number of Pages: 
Institute of Mathematical Statistics Textbooks 1
[Reviewed by
Miklós Bóna
, on

This is a book in the series of Institute of Mathematical Statistics Textbooks, and that is hard to miss. The language, the motivation, and the style of exposition is decidedly different from what this reviewer has seen, even in probability textbooks.

One problem is that the author never quite states who the target audience is. The book is part of a textbook series, so some kind of students are probably in the target audience, but what prerequisites should they have completed? In the preface, we read that not much previous knowledge is assumed, then two pages later the author assumes that the readers know what a Markov chain is. Based on this, on other similar examples, and the general level of the text, we believe that an ideal student taking a course based on this book must have had, at the very least, an advanced undergraduate course on probability or statistics. Even if the student has had an introductory level graduate course on these subjects, he or she may find this book a difficult read.

The chapters could have better introductions. Each chapter starts with an abstract, which many (maybe most) readers will not understand until they read the chapter. This reviewer is interested in combinatorics, graph theory, and probability, but he did not feel the subjects discussed in the book were well-motivated. “It is useful to be able to do calculations,” says the author on page 9. The reviewer agrees, but he wanted more introduction, more motivation, more context.

On the whole, if you decide to teach a course from this book, make sure that your students are well-prepared and well-motivated.

Miklós Bóna is Professor of Mathematics at the University of Florida.

Preface; 1. Random walks on graphs; 2. Uniform spanning tree; 3. Percolation and self-avoiding walk; 4. Association and influence; 5. Further percolation; 6. Contact process; 7. Gibbs states; 8. Random-cluster model; 9. Quantum Ising model; 10. Interacting particle systems; 11. Random graphs; 12. Lorentz gas; References; Index.