Maximum entropy mixing time of circulant Markov processes

We consider both discrete-time irreducible Markov chains with circulant transition probability matrix View the MathML source and continuous-time irreducible Markov processes with circulant transition rate matrix View the MathML source. 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 View the MathML source are maximum in the interval 0≤α≤1 when α=1/2, where View the MathML source 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 View the MathML source are also maximum in the interval 0≤α≤1 when α=1/2, where View the MathML source is the time-reversed transition rate matrix.