##
**Probability on trees and networks.**
*(English)*
Zbl 1376.05002

Cambridge Series in Statistical and Probabilistic Mathematics 42. Cambridge: Cambridge University Press (ISBN 978-1-107-16015-6/hbk; 978-1-108-73272-7/pbk; 978-1-316-67281-5/ebook). xv, 699 p. (2016).

This is a monumental book covering a lot of interesting problems in discrete probability, written by two experts in the field. Discrete probability is a fascinating area with rich connections with other areas in mathematics. The authors have done a great job of providing full proofs of all main results, hence creating a self-contained reference in this area. This has not damaged readability: the book is very well written and accessible.

The covered topics include random walks and percolation, branching processes, the mass-transport technique, isoperimetric inequalities, and more advanced topics such as Hausdorff dimension, capacity, and random walks on Galton-Watson trees.

One advantage of the book is that it is freely available on the authors’ homepages. This book is a valuable reference for researchers in discrete probability and neighboring areas. Having more than 850 exercises, it is also an excellent introduction for anyone willing to enter this beautiful area. It can also be used as textbook for various graduate courses, by an instructor who knows the book well and can spend the time choosing appropriate sections to cover.

The covered topics include random walks and percolation, branching processes, the mass-transport technique, isoperimetric inequalities, and more advanced topics such as Hausdorff dimension, capacity, and random walks on Galton-Watson trees.

One advantage of the book is that it is freely available on the authors’ homepages. This book is a valuable reference for researchers in discrete probability and neighboring areas. Having more than 850 exercises, it is also an excellent introduction for anyone willing to enter this beautiful area. It can also be used as textbook for various graduate courses, by an instructor who knows the book well and can spend the time choosing appropriate sections to cover.

Reviewer: Abbas Mehrabian (Waterloo)

### MSC:

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

05C05 | Trees |

05C80 | Random graphs (graph-theoretic aspects) |

94C05 | Analytic circuit theory |

05C81 | Random walks on graphs |

05C82 | Small world graphs, complex networks (graph-theoretic aspects) |