Permuted Random Walk Exits Typically in Linear Time

Given a permutation σ of the integers {−n, −n + 1,...,n} we consider the Markov chain Xσ, which jumps from k to σ(k ± 1) equally likely if k ≠ −n,n. We prove that the expected hitting time of {−n,n} starting from any point is Θ(n) with high probability when σ 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.