On the fastest finite Markov processes

Consider a finite irreducible Markov chain with invariant probability $π$. Define its inverse communication speed as the expectation to go from x to y, when x, y are sampled independently according to $π$. In the discrete time setting and when $π$ is the uniform distribution $\upsilon$, Litvak and Ejov have shown that the permutation matrices associated to Hamiltonian cycles are the fastest Markov chains. Here we prove (A) that the above optimality is with respect to all processes compatible with a fixed graph of permitted transitions (assuming that it does contain a Hamiltonian cycle), not only the Markov chains, and, (B) that this result admits a natural extension in both discrete and continuous time when $π$ is close to $\upsilon$: the fastest Markov chains/processes are those moving successively on the points of a Hamiltonian cycle, with transition probabilities/jump rates dictated by $π$. Nevertheless, the claim is no longer true when $π$ is significantly different from $\upsilon$.

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