Maximum entropy mixing time of circulant Markov processes

We consider both discrete-time irreducible Markov chains with circulant transition probability matrix P and continuous-time irreducible Markov processes with circulant transition rate matrix Q. In both cases we provide an expression of all the moments of the mixing time. In the discrete case, we prove that all the moments of the mixing time associated with the transition probability matrix αP+[1−α]P∗ are maximum in the interval 0≤α≤1 when α=1/2, where P∗ is the transition probability matrix of the time-reversed chain. Similarly, in the continuous case, we show that all the moments of the mixing time associated with the transition rate matrix αQ+[1−α]Q∗ are also maximum in the interval 0≤α≤1 when α=1/2, where Q∗ is the time-reversed transition rate matrix.