Random Walks on Infinite Graphs and Groups: An introduction to topological boundary theory

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Transition Probabilities for Symmetric Jump Processes (2002, background): “For a recent and comprehensive account, see the book [ Wo ].”
Random Walks on Sierpiński Graphs: Hyperbolicity and Stochastic Homogenization (2003, background): “Moreover, almost all sample paths of the Markov chain converge in the Martin compactification, and for any x ∈ X the measure ν(1) x is the hitting measure on the Martin boundary corresponding to the starting point x For a more detailed discussion of the theory of boundaries of Markov chains on graph”
A proof of Alon's second eigenvalue conjecture and related problems (2004, background): “example, page 9 of [ Woe00 ], and the theorems on λ1 in Section 3 here).”