# Mixing time of the adjacent walk on the simplex

## TL;DR

This paper analyzes the mixing time of the adjacent walk on the simplex, showing a cutoff phenomenon with a precise spectral gap and extending results to log-concave distributions.

## Contribution

It determines the spectral gap and mixing times for the adjacent walk on the simplex, revealing a cutoff phenomenon and extending to log-concave Beta distributions.

## Key findings

- Spectral gap of the adjacent walk is explicitly determined.
- Both total variation and separation distances exhibit cutoff.
- Mixing times differ by a factor of 2.

## Abstract

By viewing the $N$-simplex as the set of positions of $N-1$ ordered particles on the unit interval, the adjacent walk is the continuous time Markov chain obtained by updating independently at rate 1 the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and prove that both the total variation distance and the separation distance to the uniform distribution exhibit a cutoff phenomenon, with mixing times that differ by a factor $2$. The results are extended to the family of log-concave distributions obtained by replacing the uniform sampling by a symmetric log-concave Beta distribution.