# Analysis, Geometry, and Dynamical Systems Seminar

### Mixing time of the adjacent walk on the simplex

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 mixing time and show that the total variation distance to the uniform distribution displays a cutoff phenomenon. The results are extended to a family of log-concave distributions obtained by replacing the uniform sampling by a symmetric Beta distribution. This is joint work with Cyril Labbe' and Hubert Lacoin.