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Random Walks on Infinite Graphs and Groups

$61.00 ( ) USD

Part of Cambridge Tracts in Mathematics

  • Date Published: February 2011
  • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • format: Adobe eBook Reader
  • isbn: 9780511835469

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About the Authors
  • This eminent work focuses on the interplay between the behavior of random walks and discrete structure theory. Wolfgang Woess considers Markov chains whose state space is equipped with the structure of an infinite, locally-finite graph, or of a finitely generated group. He assumes the transition probabilities are adapted to the underlying structure in some way that must be specified precisely in each case. He also explores the impact the particular type of structure has on various aspects of the behavior of the random walk. In addition, the author shows how random walks are useful tools for classifying, or at least describing, the structure of graphs and groups.

    • 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

    "The organization of the book is well-thought-out...The reviewer has a very high opition of this book" Bulletin of the American Mathematical Society

    "a very valuable addition to the literture on this fascinating and important subject." Mathematical Review

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    Product details

    • Date Published: February 2011
    • format: Adobe eBook Reader
    • isbn: 9780511835469
    • contains: 12 b/w illus.
    • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • 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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