# Analysis of a non-reversible Markov chain speedup by a single edge

## TL;DR

This paper demonstrates that adding a single edge and introducing non-reversibility in a Markov chain can significantly accelerate mixing times, achieving an $O(n^{3/2})$ bound, unlike the $Ω(n^2)$ bound with only one modification.

## Contribution

It provides a novel example showing how combined non-reversibility and a single edge addition improve Markov chain mixing times.

## Key findings

- Adding an edge and non-reversibility reduces mixing time to $O(n^{3/2})$
- Single modifications alone do not improve mixing time beyond $Ω(n^2)$
- The combined approach yields a probabilistic improvement in mixing efficiency.

## Abstract

We present a Markov chain example where non-reversibility and an added edge jointly improve mixing time: when a random edge is added to a cycle of $n$ vertices and a Markov chain with a drift is introduced, we get mixing time of $O(n^{3/2})$ with probability bounded away from 0. If only one of the two modifications were performed, the mixing time would stay $\Omega(n^2)$.