Lower bounds for covering times for reversible Markov chains and random walks on graphs

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Pufferfish Privacy Mechanisms for Correlated Data (2016, methodology): “For a Markov Chain with transition matrix P θ , we use π θ to denote its stationary distribution [1].”
Random Walks on Graphs: A Survey (2001, background): “Aldous [1] proved that this is true up to a constant factor if the starting point is drawn at random, from the stationary distribution.”
Semisupervised alignment of manifolds (2005, background): “Correspondences may also be obtained from hand-labeled matches among samples in different data sets.”
Cutoff on all Ramanujan graphs (2015, background): “ 2d 2 log(d 1), so f(2) = 0 whereas f0(d) >0 for all d>1. Finally, when Gis Ramanujan, the sought upper bound on the L2-distance follows from considering the spectral representation (see, e.g., [3]) kPt(x;)=ˇ 1k2 L2(ˇ) = n Xn i=2 jf i(x)j2( i=d)2t (2.8) for ff ign i=1 an orthonormal basis of e”
PageRank beyond the Web (2014, background): “…unique stationary distribution [Stewart, 1994, is a good numerical treatment of such issues]: This works because both the row and column sums of A and A T are identical, and the resulting construction is a reversible Markov chain [Aldous and Fill, 2002, is a good reference on this topic].”
Random walks and diffusion on networks (2016, background): “This property, called “detailed balance” in statistical physics [121] and “time reversibility” in mathematics [1, 42], does not generally hold for directed networks.”