MIXING TIMES FOR THE SIMPLE EXCLUSION PROCESS WITH OPEN BOUNDARIES

We study mixing times of the symmetric and asymmetric simple exclu-sion process on the segment where particles are allowed to enter and exit at the endpoints. We consider different regimes depending on the entering and exiting rates as well as on the rates in the bulk, and show that the pro-cess exhibits pre-cutoff and in some cases cutoff. Our main contribution is to study mixing times for the asymmetric simple exclusion process with open boundaries. We show that the order of the mixing time can be linear or ex-ponential in the size of the segment depending on the choice of the boundary parameters, proving a strikingly different (and richer) behavior for the simple exclusion process with open boundaries than for the process on the closed segment. Our arguments combine coupling, second class particle and censor-ing techniques with current estimates. A novel idea is the use of multi-species particle arguments, where the particles only obey a partial ordering.

Automated summary

In this study, the mixing times of the symmetric and asymmetric simple exclusion process are examined when particles are allowed to enter and exit at the endpoints. Depending on the rates in the bulk and the entering and exiting rates, the process exhibits pre-cutoff and cutoff behaviors. The main contribution of this study is investigating the mixing times for the asymmetric simple exclusion process with open boundaries. It is shown that the mixing time can have a linear or exponential relationship with the size of the segment, depending on the choice of boundary parameters.