On eigenfunctions of Markov processes on trees

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Multiway Spectral Partitioning and Higher-Order Cheeger Inequalities (2014, background): “In fact, in the same paper that Miclo conjectured the validity of Theorem 1.1, he conjectured that .nding such a family {.i }should be possible [Miclo 2008; Daneshgar et al. 2012].”
On hyperboundedness and spectrum of Markov operators (2014, background): “In [24] (see also [25]), we made the conjecture that there exists a mapping c : N → R∗+ such that for all (S, μ, L) as above, c(n) ι(2)n |L| ≤ λn ≤ ιn (6)”
On Times to Quasi-stationarity for Birth and Death Processes (2009, background): “Essentially, this comes from the fact that the quasi-stationary distribution ρ considered in section 3 is decreasing on J0, N − 1K (see for instance Miclo [17]) and the iterative definitions of the Markov kernels used to intertwine the previous generators.”