Selection of the carbon-monoxide poisoning phase
Each site of $\mathbb{Z}^2$ is in state $0$ (vacant), $1$ (a carbon monoxide molecule) or $2$ (an oxygen atom). A vacant site becomes $1$ at rate $p$. Each unordered nearest-neighbour pair in state $(0,0)$ changes to $(2,2)$ at rate $q/4$, each nearest-neighbour pair with one site in state $1$ and the other in state $2$ changes to $(0,0)$ at rate $r/4$, and each nearest-neighbour pair exchanges its states at rate $\gamma$. Assume $p > q > 0$, $0 < r < \infty$ and $0 < \gamma < \infty$. This is the surface-reaction model of Ziff, Gulari and Barshad, with a finite reaction rate and stirring, as studied by Durrett and Swindle; the configurations with every site in state $1$, or every site in state $2$, are absorbing (a poisoned surface). In simulations of Ziff, Gulari and Barshad, started from the all-vacant configuration with instantaneous reaction, the surface is poisoned by carbon monoxide when $p$ is large. Durrett and Swindle proved that if $p > q$ and the initial law is translation invariant, then for every finite $K \subset \mathbb{Z}^2$, $$P(\eta_t \equiv 1 \text{ on } K) + P(\eta_t \equiv 2 \text{ on } K) \to 1 \qquad (t \to \infty).$$
Problem. Let the initial states $\eta_0(x)$, $x \in \mathbb{Z}^2$, be independent and identically distributed with $P(\eta_0(0) = 1) > 0$. For every such initial law and every choice of the parameters above, is it true that, for every finite $K \subset \mathbb{Z}^2$, $$\lim_{t \to \infty} P\bigl(\eta_t(x) = 1 \text{ for all } x \in K\bigr) = 1?$$
Source: Durrett and Swindle (1994), the conjecture following Theorem 1; repeated as Open Problem 5.4.6 in Durrett's book manuscript Interacting Particle Systems (2026). Stated by AIM — Open Applied Problems for d = 2, i.i.d. initial states with a positive density of carbon monoxide, finite reaction rate and positive stirring.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- absorbing states, AIM Open Applied Problems, catalytic surface reactions, interacting particle systems
References
@misc{durrett2026ips,
author = {Durrett, Rick},
title = {Interacting Particle Systems: Ideas, Techniques, Applications},
howpublished = {Book manuscript},
year = {2026},
url = {https://sites.math.duke.edu/~rtd/PASTA/PASTA0902.pdf},
note = {Version of 2 September 2026, Section 5.4},
}
@article{durrett1994,
author = {Durrett, Rick and Swindle, Glen},
title = {Coexistence results for catalysts},
journal = {Probab. Theory Related Fields},
volume = {98},
number = {4},
year = {1994},
pages = {489--515},
doi = {10.1007/BF01192836},
}
@article{durrett2009,
author = {Durrett, Rick},
title = {Coexistence in stochastic spatial models},
journal = {Ann. Appl. Probab.},
volume = {19},
number = {2},
year = {2009},
pages = {477--496},
doi = {10.1214/08-AAP590},
eprint = {0906.2293},
archivePrefix = {arXiv},
}
@article{ziff1986,
author = {Ziff, Robert M. and Gulari, E. and Barshad, Yoav},
title = {Kinetic phase transitions in an irreversible surface-reaction model},
journal = {Phys. Rev. Lett.},
volume = {56},
number = {24},
year = {1986},
pages = {2553--2556},
doi = {10.1103/PhysRevLett.56.2553},
}
@misc{aim-catalytic-carbon-monoxide-poisoning,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Selection of the carbon-monoxide poisoning phase}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/185-catalytic-carbon-monoxide-poisoning.md},
note = {Entry 185 at commit a153f46, 2026-10-13},
}
Resolution claims
We consider a catalytic surface model on \(\Z^2\) with CO adsorption, diatomic oxygen adsorption, finite-rate reaction and finite-rate stirring. If the CO arrival rate \(p\) exceeds the oxygen parameter \(q\), we prove that every site is eventually occupied by CO almost surely, from every iid initial law with positive CO density. This gives CO poisoning on every finite set. The …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | CO phase selection for a stirred catalytic surface model v1 recorded from AIM @ 24edaad, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00413, "Selection of the carbon-monoxide poisoning phase", clopen.solutions.