  We compute the mixing rate of a non-backtracking random walk on a regular
expander. Using some properties of Chebyshev polynomials of the second kind, we
show that this rate may be up to twice as fast as the mixing rate of the simple
random walk. The closer the expander is to a Ramanujan graph, the higher the
ratio between the above two mixing rates is.
  As an application, we show that if $G$ is a high-girth regular expander on
$n$ vertices, then a typical non-backtracking random walk of length $n$ on $G$
does not visit a vertex more than $(1+o(1))\frac{\log n}{\log\log n}$ times,
and this result is tight. In this sense, the multi-set of visited vertices is
analogous to the result of throwing $n$ balls to $n$ bins uniformly, in
contrast to the simple random walk on $G$, which almost surely visits some
vertex $\Omega(\log n)$ times.
