General mixing time bounds for finite Markov chains via the absolute spectral gap

Understand

We prove an upper bound on the total variation mixing time of a finite Markov chain in terms of the absolute spectral gap and the number of elements in the state space.

• Unlike results requiring reversibility or irreducibility, this bound is finite whenever the chain converges.
• The dependence on the number of elements means that the new bound cannot capture the behavior of rapidly mixing chains; but an example shows that such dependence is necessary.
• We also provide a sharpened result for reversible chains that are not necessarily irreducible.