Relative mixing speed of ASEP and homogeneous PushTASEP

Determine which of the n-species ASEP and the homogeneous n-species t-PushTASEP converges faster to their common stationary distribution when both processes start from the same initial condition.

Background

The paper proves that the homogeneous n-species t-PushTASEP and the n-species ASEP commute and consequently share the same eigenstates. It also recalls that the two processes share the same stationary distribution.

This common stationary behavior raises a comparative question about their transient dynamics: the processes have different transition mechanisms, with ASEP involving nearest-neighbor exchanges and PushTASEP allowing long-range pushed movements. The paper does not resolve which process reaches the shared stationary distribution more rapidly from a common initial state.

References

It also gives rise to an interesting question; which one among the ASEP and the homogeneous PushTASEP mixes faster, i.e. converges faster to their common stationary distribution starting from the same initial condition.

— Multispecies inhomogeneous $t$-PushTASEP from antisymmetric fusion  (2503.00829 - Ayyer et al., 2 Mar 2025) in Section 1, page 2