Permuted Random Walk Exits Typically in Linear Time

TL;DR

This paper proves that for a random permutation, the expected time for a specific Markov chain to reach boundary points is linear in size, using properties of Eulerian expanders.

Contribution

It establishes that permuted random walks on integers typically have linear hitting times, leveraging the structure of Eulerian expanders in the analysis.

Findings

Expected hitting time is Theta(n) for random permutations.

The digraph of transitions forms an Eulerian expander with high probability.

Hitting times are estimated using properties of directed Eulerian expanders.

Abstract

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.