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 \(\tau \) 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 \(\tau _\textsf {RW}\) of the lazy random walk on G is characterised by the edge conductance \(\Phi \) of G via Cheeger’s inequality: \(\Phi ^{-1} \lesssim \tau _\textsf {RW} \lesssim \Phi ^{-2} \log |V|\). Analogously, we characterise the fastest mixing time \(\tau ^\star \) via a Cheeger-type inequality but for a different geometric quantity, namely the vertex conductance \(\Psi \) of G: \(\Psi ^{-1} \lesssim \tau ^\star \lesssim \Psi ^{-2} (\log |V|)^2\). 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 \(\varepsilon \)-close to uniform in total variation. We show that it is always possible to construct such a chain with mixing time \(\tau \lesssim \varepsilon ^{-1} ({\text {diam}} G)^2 \log |V|\). Finally, we discuss analogous questions for continuous-time and time-inhomogeneous chains.