Existence of a single sharp dynamical critical density

Determine whether the rapid-absorption and prolonged-transience regimes for the higher-dimensional facilitated exclusion process are separated by a single sharp dynamical critical density.

Background

The paper proves two separated dynamical regimes for the facilitated exclusion process in dimensions d≥2: poly-logarithmic absorption times at sufficiently low Bernoulli particle densities and a lower bound of order N/log N for the transience time when the density lies in (1/2,3/4). These results do not identify a critical density separating the regimes or establish that such a threshold is unique and sharp.

The authors explicitly ask whether the observed separation is governed by one dynamical critical density, in analogy with absorbing-state phase transitions in other conservative particle systems.

References

Determining whether they are separated by a single sharp dynamical critical density remains an open problem.

— Facilitated Exclusion Process in Higher Dimensions: Recurrent Structure and Transient Dynamics  (2609.18885 - Kim et al., 16 Sep 2026) in Section 1, subsection “Our Achievements,” final paragraph before Section 2

The optimal transience-time scales and the recurrent classes selected by Bernoulli initial data outside the low-density regime also remain to be understood.

— Facilitated Exclusion Process in Higher Dimensions: Recurrent Structure and Transient Dynamics  (2609.18885 - Kim et al., 16 Sep 2026) in Section 1, subsection “Our Achievements,” final paragraph before Section 2

We conjecture that, for every $d \ge 2$ and $\rho \in (1/2 , \rho_\star{\rm st} )$, there exists a constant $c=c(d,\rho)>0$ such that \begin{equation}\label{eq:conjecture} \lim_{N\to\infty} \mathbb{P}{N}{\nu{\rho}\left[\tau_{\rm tr} > e{cNd} \right]=1. \end{equation}

eq:conjecture:

lim⁡N→∞PνρN[τtr>ecNd]=1.\lim_{N\to\infty} \mathbb{P}^{N}_{\nu_{\rho}}\left[\tau_{{\rm tr}} > e^{cN^d} \right]=1.

— Facilitated Exclusion Process in Higher Dimensions: Recurrent Structure and Transient Dynamics  (2609.18885 - Kim et al., 16 Sep 2026) in Remark following Theorem 3 in Section 2, subsection “Dynamical Regimes for Higher-Dimensional FEPs”