The following is a quote from Lifting Markov Chains to Speed up Mixing, by Chen, Lovasz, and Pak:
...Thus we have described a (randomized) stopping rule that, for any starting node, stops in an expected number $O(\sqrt{n})$ steps, and the probability of stopping at each node is at least 1/2 its stationary probability. By standard results (see e.g. [9]), this implies that the mixing time is $O(\sqrt{n})$.
My question: I would like a precise reference for this fact, especially one which has a proof I can read. The reference [9] is Mixing Times by Lovasz and Winkler, and I was unable to find this statement in there.
Moreover, there is some ambiguity of what a mixing time of means; what I want it to mean in this case is that except from the eigenvalue at $1$, every other eigenvalue of the chain has modulus at most $1-c/\sqrt{n}$ for some constant $c>0$.
Can someone provide a precise reference?