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.