By viewing the @math -simplex as the set of positions of @math 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 @math . The results are extended to the family of log-concave distributions obtained by replacing the uniform sampling by a symmetric log-concave Beta distribution.
We give a new and elementary computation of the spectral gap of the Kac walk on S N. The result is obtained as a by-product of a more general obser- vation which allows to reduce the analysis of the spectral gap of an N-component system to that of the same system for N = 3. The method applies to a number of random "binary collision" processes with complete-graph structure, including non-homogeneous examples such as exclusion and colored exclusion processes with site disorder.