Spectral gap of Markov chains on a cycle

We search for the Markov chain with the optimal mixing rate where transitions are restricted to happen along a cycle of the states. We show that homogeneous, reversible chains are locally optimal for perturbations that make them inhomogeneous and non-reversible. Moreover, we show the optimality holds globally if only a single type of perturbation (either inhomogeneous or non-reversible) is applied. However, we conjecture global optimality holds for mixed perturbations as well, which is backed by simulation results. This paper complements previous results on bounds for mixing times of general Markov chains on the cycle [1].