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.
Let us briefly describe the main arguments used to establish the results of this article. Concerning the spectrum, the first observation is that when restricted to the invariant subspace of linear functions, the generator can be easily diagonalised. In particular, one finds that @math is an eigenfunction of @math with eigenvalue @math and therefore, for all @math , the spectral gap is at most @math . Proving that this is actually the spectral gap is more delicate. We prove it only in the case @math since we rely on the FKG inequality. In this case, using the strict monotonicity of @math , we show that any eigenfunction associated to an eigenvalue larger than @math needs to be non-decreasing as well, and the FKG inequality is then applied to conclude that the only such eigenfunction is the constant one; see Section below. Concerning the case @math , we can only assert that the spectral gap is not smaller than @math for some constant @math . Indeed, this is a direct consequence of the mean field spectral gap and a comparison argument (see e.g. @cite_18 ).
@cite_18
2112888650
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.