Mixing on the cycle with constant size perturbation

TL;DR

This paper demonstrates that introducing a small, constant-size perturbation to a cycle Markov chain with non-reversible transitions significantly improves mixing times, achieving near-quadratic speedup.

Contribution

It provides a novel construction showing how adding a fixed number of random edges to a cycle enhances mixing times with non-reversible Markov chains.

Findings

Mixing time of $O(n^{(k+2)/(k+1)})$ achieved with $k$ added edges.

Near-quadratic improvement over standard cycle mixing.

Construction based on biased random walk along the cycle.

Abstract

Considering a Markov chain defined on a cycle, near-quadratic improvement of mixing is shown when only a subtle perturbation is introduced to the structure and non-reversible transition probabilities are used. More precisely, a mixing time of $O(n^{\frac{k+2}{k+1}})$ can be achieved by adding $k$ random edges to the cycle, keeping $k$ fixed while $n\to\infty$. The construction builds upon a biased random walk along the cycle.