Mixing-time and cutoff conjecture for open asymmetric simple exclusion processes

Let $N$ be the size of a segment on which the asymmetric simple exclusion process evolves with open boundaries. Its parameter space has a high-density phase, a low-density phase, a maximum-current phase, and a coexistence line.

Mixing-time and cutoff conjecture. The following claims should hold:

1. In the high- and low-density phases, the mixing time is of order $N$, and cutoff occurs.
2. In the maximum-current phase, the mixing time is of order $N^{3/2}$, and no cutoff occurs.
3. On the coexistence line, the mixing time is of order $N^2$, and no cutoff occurs.

The paper establishes a full characterization for the totally asymmetric simple exclusion process with open boundaries, whereas sharp mixing-time estimates and cutoff results for the asymmetric simple exclusion process remain open. Existing results provide only an order-$N$ bound in the high- and low-density phases and an order-$N^3$ bound at the triple point.