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Probability on Trees and Networks
About this book
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.
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Information
Table of contents
- Cover
- Half title
- Dedication
- Series
- Title
- Copyright
- Contents
- Preface
- Chapter 1: Some Highlights
- Chapter 2: RandomWalks and Electric Networks
- Chapter 3: Special Networks
- Chapter 4: Uniform Spanning Trees
- Chapter 5: Branching Processes, Second Moments, and Percolation
- Chapter 6: Isoperimetric Inequalities
- Chapter 7: Percolation on Transitive Graphs
- Chapter 8: The Mass-Transport Technique and Percolation
- Chapter 9: Infinite Electrical Networks and Dirichlet Functions
- Chapter 10: Uniform Spanning Forests
- Chapter 11: Minimal Spanning Forests
- Chapter 12: Limit Theorems for Galton-Watson Processes
- Chapter 13: Escape Rate of RandomWalks and Embeddings
- Chapter 14: RandomWalks on Groups and Poisson Boundaries
- Chapter 15: Hausdorff Dimension
- Chapter 16: Capacity and Stochastic Processes
- Chapter 17: RandomWalks on Galton-Watson Trees
- Comments on Exercises
- Bibliography
- Glossary of Notation
- Index