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9780521061728

Random Walks on Infinite Graphs and Groups

by
  • ISBN13:

    9780521061728

  • ISBN10:

    0521061725

  • Edition: 1st
  • Format: Paperback
  • Copyright: 2008-05-19
  • Publisher: Cambridge University Press

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Summary

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.

Table of Contents

The Type Problem
Basic facts
Recurrence and transience of infinite networks
Applications to random walks
Isoperimetric inequalities
Transient subtrees, and the classification of the recurrent quasi transitive graphs
More on recurrence
The Spectral Radius
Superharmonic functions and r-recurrence
The spectral radius
Computing the Green function
Spectral radius and strong isoperimetric inequality
A lower bound for simple random walk
Spectral radius and amenability
The Asymptotic Behaviour of Transition Probabilities
The local central limit theorem on the grid
Growth, isoperimetric inequalities, and the asymptotic type of random walk
The asymptotic type of random walk on amenable groups
Simple random walk on the Sierpinski graphs
Local limit theorems on free products
Intermezzo
Free groups and homogenous trees
An Introduction to Topological Boundary Theory
Probabilistic approach to the Dirichlet problem, and a class of compactifications
Ends of graphs and the Dirichlet problem
Hyperbolic groups and graphs
The Dirichlet problem for circle packing graphs
The construction of the Martin boundary
Generalized lattices, Abelian and nilpotent groups, and graphs with polynomial growth
The Martin boundary of hyperbolic graphs
Cartesian products
Table of Contents provided by Publisher. All Rights Reserved.

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