# Mixing time of an unaligned Gibbs sampler on the square

## TL;DR

This paper proves that the mixing time of a specific unaligned Gibbs sampler on the unit square is proportional to the square of a parameter, confirming a conjecture and improving previous exponential bounds.

## Contribution

The paper confirms Diaconis's conjecture that the mixing time is O(A^2) for the unaligned Gibbs sampler, improving the previous exponential estimate.

## Key findings

- Mixing time is proportional to A^2.
- Confirmed conjecture by Diaconis.
- Improved from exponential to quadratic bound.

## Abstract

The paper concerns a particular example of the Gibbs sampler and its mixing efficiency. Coordinates of a point are rerandomized in the unit square $[0,1]^2$ to approach a stationary distribution with density proportional to $\exp(-A^2(u-v)^2)$ for $(u,v)\in [0,1]^2$ with some large parameter $A$. Diaconis conjectured the mixing time of this process to be $O(A^2)$ which we confirm in this paper. This improves on the currently known $O(\exp(A^2))$ estimate.