Low-Vacancy Diffusion in Kinetically Constrained Lattice Gases

Definitions and mathematical statement

Let $q$ denote the vacancy density and let $D(q)$ denote the diffusion coefficient referred to in the supplied kinetically constrained lattice-gas problem. The low-vacancy limit is $q\downarrow0$ through positive densities.

The requested problem is to determine the correct asymptotic order of $D(q)$ in this limit. If "asymptotic order" means comparison up to positive multiplicative constants, the target would be to identify a positive function $f$ and prove that there exist constants $c,C,q_0>0$, independent of $q$, such that \[ c f(q)\leq D(q)\leq C f(q) \qquad\text{for every admissible }q\text{ with }0<q<q_0. \] This interpretation is conditional: the source does not specify whether multiplicative comparison, logarithmic asymptotics, or an asymptotic equivalent is intended.

The supplied record does not specify the lattice, dimension, kinetic constraint, transition rates, equilibrium law, or definition of the diffusion coefficient. In particular, it does not identify tagged-particle versus collective diffusion, or whether $D(q)$ is a scalar or a diffusion matrix. Thus the notation above records the intended question but does not define a unique mathematical model or a fully determined asymptotic problem. Any fixed model parameters and their role in the comparison constants must be specified before the displayed target can be made precise.

The supplied dated assessment states that diffusion is positive at every positive vacancy density but that its correct low-vacancy asymptotic order remains undetermined. This is a supplied literature claim with unspecified model scope, not an independently verified theorem or status assessment.

Short English statement

Determine how the diffusion coefficient behaves as the vacancy density tends to zero in the intended kinetically constrained lattice-gas model. The model, the meaning of diffusion, and the required precision of the asymptotics are unspecified, so a unique precise formulation requires further source clarification. The supplied open label and positivity claim are dated curation.

Sources

• Local UnsolvedMath registry, permanent ID $30003284$, source problem code catalogue entry, title: Low-Vacancy Diffusion in Kinetically Constrained Lattice Gases; collection: Oberwolfach Reports Open Problems. The record gives proposed year 2016 and identifies no proposer.
• Supplied report citation: Large Scale Stochastic Dynamics (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/54, pp. 3031--3086. https://doi.org/10.4171/owr/2016/54. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The literature-source label for this same link is Oberwolfach Report 54 (2016); the differing year descriptions are retained without resolution.
• Supplied literature source: Blondel and Toninelli. https://arxiv.org/abs/1701.07663. The record supplies no title, theorem number, or passage identifying the precise model and diffusion coefficient relevant here.
• Literature assessment checked at 2026-08-21: the supplied status is open, with the assessment that positive diffusion at every positive vacancy density is proved while the correct low-vacancy asymptotic order remains undetermined. This is supplied dated curation derived from the OWR material, not independently verified here; time-sensitive status and cited claims require independent verification.