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Random Walks on Infinite Graphs and Groups


Random Walks on Infinite Graphs and Groups Cover


Synopses & Reviews

Publisher Comments:

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.


The main theme of this book is the interplay between random walks and discrete structure theory.

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.

Product Details

Woess, Wolfgang
Cambridge University Press
Wolfgang, Woess
Probability & Statistics - General
Differential Equations
Mathematics | Probability and Statistics
Edition Description:
Trade paper
Cambridge Tracts in Mathematics
Series Volume:
Publication Date:
12 b/w illus.
9.00x6.00x.79 in. 1.14 lbs.

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Related Subjects

History and Social Science » World History » General
Science and Mathematics » Mathematics » Differential Equations
Science and Mathematics » Mathematics » Probability and Statistics » General
Science and Mathematics » Mathematics » Probability and Statistics » Statistics

Random Walks on Infinite Graphs and Groups New Trade Paper
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Product details 352 pages Cambridge University Press - English 9780521061728 Reviews:
"Synopsis" by , The main theme of this book is the interplay between random walks and discrete structure theory.
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