  We study regular graphs in which the random walks starting from a positive
fraction of vertices have small mixing time. We prove that any such graph is
virtually an expander and has no small separator. This answers a question of
Pak [SODA, 2002]. As a corollary, it shows that sparse (constant degree)
regular graphs with many well-mixing vertices have a long cycle, improving a
result of Pak. Furthermore, such cycle can be found in polynomial time.
  Secondly, we show that if the random walks from a positive fraction of
vertices are well-mixing, then the random walks from almost all vertices are
well-mixing (with a slightly worse mixing time).
