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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.

Random walks (Mathematics) --- Graph theory. --- Infinite groups.

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Lie, Algèbres de, de dimension infinie. --- Infinite dimensional Lie algebras.

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Topology --- 512.73 --- Infinite-dimensional manifolds --- K-theory --- Algebraic topology --- Homology theory --- Manifolds, Infinite-dimensional --- Global analysis (Mathematics) --- Topological manifolds --- Cohomology theory of algebraic varieties and schemes --- 512.73 Cohomology theory of algebraic varieties and schemes --- Infinite-dimensional manifolds. --- K-theory. --- K-théorie --- K-théorie.

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Operator theory --- 517.986 --- Hilbert space --- Morita duality --- Operator algebras --- Algebras, Operator --- Topological algebras --- Duality, Morita --- Morita's duality --- Modules (Algebra) --- Banach spaces --- Hyperspace --- Inner product spaces --- Topological algebras. Theory of infinite-dimensional representations --- 517.986 Topological algebras. Theory of infinite-dimensional representations --- Hilbert space. --- Morita duality. --- Hilbert, Espaces de --- Hilbert, Espaces de.

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Quantum groups --- 517.986 --- Enveloping algebras, Quantized --- Function algebras, Quantized --- Groups, Quantum --- Quantized enveloping algebras --- Quantized function algebras --- Quantum algebras --- Group theory --- Mathematical physics --- Quantum field theory --- 517.986 Topological algebras. Theory of infinite-dimensional representations --- Topological algebras. Theory of infinite-dimensional representations --- Quantum theory --- Théorie des groupes --- Théorie quantique