Geometric bounds on the fastest mixing Markov chain.

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 RW of the lazy random walk on G is characterised by the edge conductance Φ of G via Cheeger's inequality: Φ - 1 ≲ τ RW ≲ Φ - 2 log | 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. [ABSTRACT FROM AUTHOR]