**Update, July 2011.**
The conjecture has been solved in this paper by Aaron Smith.
## Mixing time for a Gibbs sampler on the simplex

Fix $d$ and consider the simplex
$\Delta = \{ (x_1,\ldots,x_d) : x_i \geq 0, \sum_i x_i = 1\}$.
Consider the discrete-time Markov chain on $\Delta$ with steps:
from state (x_1,\ldots,x_d), pick 2 distinct coordinates
$i,j$ uniformly at random, and replace the 2 entries
$x_i,x_j$ by
$U, x_i+x_j - U$
where $U$ is uniform on $(0,x_i+x_j)$.

The stationary distribution $\pi$ is the uniform distribution on $\Delta$.
A coupling argument given in
Chapter 13 section 1.4 of the draft Aldous-Fill book
shows that the mixing time is at most order $d^2 \log d$.

But by analogy with the "card shuffling by random transposition" model
we conjecture that in fact the mixing time is order $d \log d$.

**Problem:** Prove this conjecture, e.g. by using ideas from the
recent Burton - Kovchegov coupling proof of the card-shuffling bound.