RANDOM WALKS ON INFINITE GRAPHS AND GROUPS (Cambridge Tracts in Mathematics 138)

By L. Saloff-Coste. 2001

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Connected Lie groups and property RD (2004, background): “The authors observed [4] that property RD is also relevant to the study of the asymptotic behavior of the probability of return of random walks on non-amenable groups (for a general reference on random walks, see [36]).”
Random walks on graphs: ideas, techniques and results (2005, methodology): “Using combinatorial techniques analogous to those presented for bundled graphs, it can be shown that the local and the average spectral dimensions on the whole graph are the sums of the corresponding dimensions of the two initial graphs [7].”
Analysis of the physical Laplacian and the heat flow on a locally finite graph (2008, background): “For related topics as random walks and analysis on networks, we refer to [16, 21, 22] and the references therein.”
A Dirichlet principle for non reversible Markov chains and some recurrence theorems (2011, methodology): “One can use the Dirichlet principle to prove the recurrence of random walks in random conductances in dimension one and two, and the Thomson principle to prove transience in dimension larger than or equal to three [15, 17].”
Evolving sets, mixing and heat kernel bounds (2003, unknown): “Similarly, for y ∈ Sc we have Q(S, y) ∧Q(Sc, y) ≥ γ1−γQ(S, y).”