Dice Question Streamline Icon: https://streamlinehq.com

Linear mixing-time scaling in higher dimensions for FA-1f on finite boxes

Determine whether, for FA-1f on the finite box \mathbb{Z}^d_n with infected boundary conditions ensuring irreducibility, the mixing time scales linearly with n for all infection densities q > 0.

Information Square Streamline Icon: https://streamlinehq.com

Background

While linear mixing-time scaling is known in one-dimensional FA-1f with appropriate boundary conditions, extending this to higher dimensions remains challenging. Comparisons with contact processes yield results for sufficiently large q, but small-q behavior is unresolved.

Establishing universal linear scaling would align FA-1f with heuristic expectations that facilitation propagates in a diffusive-like manner across dimensions under ergodic conditions, and it would provide quantitative control of non-equilibrium dynamics on finite domains.

References

Finally, a related question in higher dimensions is the scaling with $n$ of the mixing time of the (ergodic) chain on $\mathbb {Z} d_n$. The conjecture is a linear scaling for all values of $q>0$, but, once more, that has been proved only for $q$ large enough .

— Long time behaviour of one facilitated kinetically constrained models: results and open problems (2510.20461 - Martinelli et al., 23 Oct 2025) in Section 1.1 (State of the art and some conjectures)