We begin by introducing a non-reversible interacting particle system, the Marked Arrow Model (MAM), which can be viewed both as a lifting of a lazy totally asymmetric simple exclusion process (TASEP) and as a zero-temperature true self-avoiding walk (TSAW) in a periodic setting. A key question for MAM is how the relevant time scales for relaxation to equilibrium on the discrete periodic ring depend on the number of sites, L. 

We study two different types of mixing times: the usual total variation distance mixing time and the T-averaged mixing time, controlling the convergence of the law of the process averaged over the last T steps. While they scale both diffusively, interestingly, while the latter exhibits cutoff, the former does not.  On the way, we will also explain the connections and differences among the two different notions of mixing times and we will explain how our results allow us to prove a conjecture formulated by physicists on a model that has been put forward as a paradigmatic example of fast non-reversible sampling Monte Carlo algorithms.