Mixing times for the TASEP in the maximal current phase

Abstract

We study mixing times for the totally asymmetric simple exclusion process (TASEP) on a segment of size N with open boundaries. We focus on the maximal current phase, and prove that the mixing time is of order N 3/2 , up to logarithmic corrections. In the triple point, where the TASEP with open boundaries approaches the Uniform distribution on the state space, we show that the mixing time is precisely of order N 3/2 . This is conjectured to be the correct order of the mixing time for a wide range of particle systems with maximal current. Our arguments rely on a connection to last-passage percolation, and recent results on moderate deviations of last-passage times.

Riemann surfaces for KPZ with periodic boundaries

“…The scaling L 3/2 for the relaxation time means that spectral gaps are of order L −3/2 , see sections 3.2.4 and 2.4.3. More generally, this scaling is also observed for the mixing time, which considers the convergence of the probabilities of all the configurations to their stationary values and not just the convergence of the statistics of the height [171,172]. An intriguing synchronization behaviour for the trajectories of microstates [173] is also observed numerically on the same time scale.…”

Some recent progress on the stationary measure for the open KPZ equation

“…In the case of the triple point between all three phases, they are able to give an upper bound of order N 3 . However, the expectation is that the true mixing time at this point and in the maximal current phase is of order N 3/2 and Schmid [Sch21] has since proved this in the case of TASEP (q = 0).…”

Mixing time and cutoff for one dimensional particle systems

“…Mixing time results for the exclusion of the segment with a variety of boundary condition are proved in [15], where several open questions and conjectures are also displayed. One of these conjecture is solved in [43], where it shown that in the maximal current phase, for the totally asymmetric exclusion process (TASEP) the mixing time in that case is of order N 3/2 . A similar result is predicted to hold for the asymmetric exclusion process on the circle, and the corresponding lower bound on the mixing time can be deduced from the results in [1].…”