Probability on Trees and Networks (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 42)

Description

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike. Review: Excellent book - I am only a ‚hobby mathematician‘, so my comments may be read with this in mind. The book delivers exactly what the title promises. The authors motivate very well what they are doing, it is clear that they are masters in their field! Ther proofs are generally clear and understandable without to many ‚side calculations‘, although towards the end of some chapters they become harder to understand, they skip steps. A good background of probability theory and of measure theory (which I do not have) is definitely helpful, while a background in grapf theory is not really needed. The book contains a huge number of exercises in each chapter (which I did not attempt to do), hints are given for some of them One drawback( and the reason I deducted one star) is the reference to exercises in many proofs - and often no hints are given even for those. This makes the book a bit harder for self study. The book contains very few typos - mostly of a harmless nature. Amazing for a first edition. All in all, I highly recommend this book to anybody interested in this subject! The ‚physical‘ quality of the book is also excellent: very clear print, good paper, the pages do not fall out even after heavy and extended use.