Cutoff for congestion dynamics and related generalized exclusion processes

We consider congestion dynamics with $n$ players and $Q$ resources under the constraint that the number of each resource is $\kappa$ and that $n<\kappa Q$ in the regime that $n$ and $\kappa$ diverge but $Q$ is fixed with $n=\lfloor{\rho \kappa Q\rfloor}$ for a fixed constant $\rho \in (0, 1/2]$. We show that the Glauber dynamics and its unlabeled version exhibit cutoff at time $(1/2)n \log n$ and $(1/2)(1-\rho)n\log n$ in total variation respectively. The unlabeled version is a special case of natural Markov chains for sampling from log M-concave distributions. We also show that a family of Markov chains for uniform sampling on M-convex sets does not necessarily exhibit cutoff.