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Random Walks on Infinite Graphs and Groups

Part of Cambridge Tracts in Mathematics

  • Date Published: May 2008
  • availability: Available
  • format: Paperback
  • isbn: 9780521061728

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About the Authors
  • The main theme of this book is the interplay between the behaviour of a class of stochastic processes (random walks) and discrete structure theory. The author considers Markov chains whose state space is equipped with the structure of an infinite, locally finite graph, or as a particular case, of a finitely generated group. The transition probabilities are assumed to be adapted to the underlying structure in some way that must be specified precisely in each case. From the probabilistic viewpoint, the question is what impact the particular type of structure has on various aspects of the behaviour of the random walk. Vice-versa, random walks may also be seen as useful tools for classifying, or at least describing the structure of graphs and groups. Links with spectral theory and discrete potential theory are also discussed. This book will be essential reading for all researchers working in stochastic process and related topics.

    • Wide-ranging treatment that connects to many other areas of mathematics
    • Author is acknowledged expert in the field
    • Exciting area
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    Reviews & endorsements

    Review of the hardback: 'This is an excellent book, where beginners and specialists alike will find useful information. It will become one of the major references for all those interested directly or indirectly in random walks. I highly recommend it.' L. Saloff-Coste, Bulletin of the London Mathematical Society

    Review of the hardback: '… will be essential reading for all researchers in stochastic processes and related topics.' European Maths Society Journal

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    Product details

    • Date Published: May 2008
    • format: Paperback
    • isbn: 9780521061728
    • length: 352 pages
    • dimensions: 229 x 152 x 20 mm
    • weight: 0.52kg
    • contains: 12 b/w illus.
    • availability: Available
  • Table of Contents

    Part I. The Type Problem:
    1. Basic facts
    2. Recurrence and transience of infinite networks
    3. Applications to random walks
    4. Isoperimetric inequalities
    5. Transient subtrees, and the classification of the recurrent quasi transitive graphs
    6. More on recurrence
    Part II. The Spectral Radius:
    7. Superharmonic functions and r-recurrence
    8. The spectral radius
    9. Computing the Green function
    10. Spectral radius and strong isoperimetric inequality
    11. A lower bound for simple random walk
    12. Spectral radius and amenability
    Part III. The Asymptotic Behaviour of Transition Probabilities:
    13. The local central limit theorem on the grid
    14. Growth, isoperimetric inequalities, and the asymptotic type of random walk
    15. The asymptotic type of random walk on amenable groups
    16. Simple random walk on the Sierpinski graphs
    17. Local limit theorems on free products
    18. Intermezzo
    19. Free groups and homogenous trees
    Part IV. An Introduction to Topological Boundary Theory:
    20. Probabilistic approach to the Dirichlet problem, and a class of compactifications
    21. Ends of graphs and the Dirichlet problem
    22. Hyperbolic groups and graphs
    23. The Dirichlet problem for circle packing graphs
    24. The construction of the Martin boundary
    25. Generalized lattices, Abelian and nilpotent groups, and graphs with polynomial growth
    27. The Martin boundary of hyperbolic graphs
    28. Cartesian products.

  • Author

    Wolfgang Woess, Technische Universität Graz, Austria

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