Geometric Bounds on the Fastest Mixing Markov Chain
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In the Fastest Mixing Markov Chain problem, we are given a graph G = (V, E) and desire the discrete-time Markov chain with smallest mixing time τ subject to having equilibrium distribution uniform on V and non-zero transition probabilities only across edges of the graph. It is well-known that the mixing time τ_ of the lazy random walk on G is characterised by the edge conductance Φ of G via Cheeger's inequality: Φ^-1≲τ_≲Φ^-2log |V|. Analogously, we characterise the fastest mixing time τ^⋆ via a Cheeger-type inequality but for a different geometric quantity, namely the vertex conductance Ψ of G: Ψ^-1≲τ^⋆≲Ψ^-2 (log |V|)^2. This characterisation forbids fast mixing for graphs with small vertex conductance. To bypass this fundamental barrier, we consider Markov chains on G with equilibrium distribution which need not be uniform, but rather only ε-close to uniform in total variation. We show that it is always possible to construct such a chain with mixing time τ≲ε^-1 (diam G)^2 log |V|. Finally, we discuss analogous questions for continuous-time and time-inhomogeneous chains.
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