  Given a permutation sigma of the integers {-n,-n+1,...,n} we consider the
Markov chain X_{sigma}, which jumps from k to sigma (k\pm 1) equally likely if
k\neq -n,n. We prove that the expected hitting time of {-n,n} starting from any
point is Theta(n) with high probability when sigma is a uniformly chosen
permutation. We prove this by showing that with high probability, the digraph
of allowed transitions is an Eulerian expander; we then utilize general
estimates of hitting times in directed Eulerian expanders.
