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Consider the biased simple random walk on the circle $\mathbb{Z}/N\mathbb{Z}$, say in continuous time to avoid periodicity where particle hops to left with probability $q$ and right with probability $p$ after a rate one exponential waiting time.

It is easy to see the unique stationary distribution is uniform distribution on the circle and when $p\neq q$, this chain is non-reversible.

My question is: is there any known result about the mixing time for this chain? In particular does it have cutoff? I could not find related results in Levin and Peres' book. Since this chain is non-reversible, I expect the spectral method would not give very good bounds on the mixing time since the eigenvalues are complex...

In general for non-reversible Markov chains to what extend do we know about their mixing times? Is there any good reference for this topic?

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