Random Walks on Infinite Graphs and Groups: The type problem

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Evolving sets, mixing and heat kernel bounds (2003, background): “For infinite Markov chains, Nash inequalities have been developed for general isoperimetric profiles; see Varopoulos [24], the survey by Pittet and Saloff Coste [21], the book [25], and especially the work of Coulhon [6, 7].”
Ends in Uniform Spanning Forests (2007, background): “Rough isometry preserves non-amenability (Woess (2000), Theorem 4.7) and Hd is non-amenable when d ≥ 2.”
Recent progress on the Random Conductance Model (2011, background): “We will not try to delve deeper into the details of historical developments of the subject; instead, the reader should consult the many texts that have been written on this (e.g., by Coulhon and Grigor’yan [36], Davies [38], Kumagai [90], Montenegro and Tetali [105], Varopoulos [131], Varopoulos, Sa”