Random Walks on Infinite Graphs and Groups
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.
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amenable analytic aperiodic apply Borel bounded geometry Cartesian product Cayley graph choose compact compute consider constant convergence Corollary deﬁne deﬁnition denote Dirichlet problem edges element end compactiﬁcation equivalent example exponential ﬁnd ﬁnite range ﬁnitely generated groups ﬁrst ﬁxed point ﬂow free group free product geodesic harmonic functions Hence homogeneous tree hyperbolic graph implies inﬁnite integral isometry isoperimetric inequality Kaimanovich kernel lattices Lemma Let F limit theorem locally ﬁnite graph Markov chain Martin boundary Martin compactiﬁcation metric minimal neighbour random walk neighbourhood non-negative obtain p-recurrent particular path probability measure proof of Theorem Proposition prove Recall recurrent respect result rough isometry roughly isometric satisﬁes sequence simple random walk spectral radius subgraph subgroup of AUT(X subset Suppose symmetric t-harmonic tiling topology transient transition matrix transition probabilities uniformly irreducible upper bound vertex degrees vertex-transitive graph vertices Woess write yields