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This introduction to some of the principal models in the theory of disordered systems leads the reader through the basics, to the very edge of contemporary research, with the minimum of technical fuss. Topics covered include random walk, percolation, self-avoiding walk, interacting particle systems, uniform spanning tree, random graphs, as well as the Ising, Potts, and random-cluster models for ferromagnetism, and the Lorentz model for motion in a random medium. Schramm-Löwner evolutions (SLE) arise in various contexts. The choice of topics is strongly motivated by modern applications and focuses on areas that merit further research. Special features include a simple account of Smirnov's proof of Cardy's formula for critical percolation, and a fairly full account of the theory of influence and sharp-thresholds. Accessible to a wide audience of mathematicians and physicists, this book can be used as a graduate course text. Each chapter ends with a range of exercises.
- Sales Rank: #1358653 in Books
- Published on: 2010-08-16
- Released on: 2010-06-24
- Original language: English
- Number of items: 1
- Dimensions: 8.98" h x .55" w x 5.98" l, .90 pounds
- Binding: Paperback
- 260 pages
Review
"The book under review serves admirably for this "getting started" purpose. It provides a rigorous introduction to a broad range of topics centered on the percolation-IPS field discussed above... This book, like a typical Part III course, requires only undergraduate background knowledge but assumes a higher level of general mathematical sophistication. It also requires active engagement by the reader. As I often tell students, "Mathematics is not a spectator sport - you learn by actually doing the exercises!" For the reader who is willing to engage the material and is not fazed by the fact that some proofs are only outlined or are omitted, this style enables the author to cover a lot of ground in 247 pages."
David Aldous, Bulletin of the American Mathematical Society
"It is written in a condensed style with only the briefest of introductions or motivations, but it is a mine of information for those who are well prepared and know how to use it. It formed the basis for a Probability reading group at the University of Warwick last term and was well received, and parts of it are being used by a colleague for an undergraduate module this term on Probability and Discrete Mathematics."
R.S. MacKay, Contemporary Physics
About the Author
Geoffrey Grimmett is Professor of Mathematical Statistics in the Statistical Laboratory at the University of Cambridge.
Most helpful customer reviews
10 of 11 people found the following review helpful.
Exactly what the title says!
By PST
This book covers exactly what the title promises: Various applications of random prozesses 'living' on graphs.
The mathematical background is not very demanding. Other than some terminology nothing is really needed from graph theory. A good background of probability theory (but not the measure theoretic part) and some knowledge of Markov processes will be helpful, but not absolutely required.
The proofs are not too hard, but at times an additional word of explanation would have rendered them a bit easier.
Unfortunately, some important proofs are given as excercises - a bad habit in my opinion, mostly if no solutions to the excercies are given.
The chapter of the "Quantum Ising Model" was incomprehensible for me, too much physics required...
A book with such a broad topic is either too big, or else is touches many details only in a very cursory manner - the latter is the case here. However, the bibliography seems excellent, making up for this lack.
The book contains virtually no typos - important for self study.
Overall, a good, solid book!
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