The density conjecture for activated random walk on the square lattice
Fix a sleep rate $\lambda > 0$. In activated random walk on $\mathbb{Z}^2$, active particles perform independent continuous-time simple random walks with jump rate $1$; an active particle alone on its site falls asleep at rate $\lambda$, and a sleeping particle becomes active when an active particle arrives at its site. Let $\zeta_c(\lambda)$ be the infimum of the densities $\zeta$ for which the system started from i.i.d. Poisson($\zeta$) active particles almost surely fails to fixate, where fixation means that only finitely many jumps occur at every site. Rolla, Sidoravicius and Zindy proved that this threshold is the same for every stationary ergodic initial configuration of active particles with mean $\zeta$.
For $\Lambda_n = [-n, n]^2 \cap \mathbb{Z}^2$, consider the driven-dissipative Markov chain on stable configurations of $\Lambda_n$ (at most one sleeping particle per site, no active particles): at each step one active particle is added at a uniformly chosen site of $\Lambda_n$ and the system runs until it is stable, particles that jump out of $\Lambda_n$ being killed. Let $\pi_n$ be its stationary law and $N(\eta)$ the number of sleeping particles in a configuration $\eta$. Hoffman, Johnson and Junge proved the one-dimensional analogue: on intervals of $\mathbb{Z}$ the expected density of sleeping particles under the stationary law converges to the fixation threshold on $\mathbb{Z}$, for every $\lambda > 0$. Forien gave another proof.
Problem. Is $$\lim_{n \to \infty} \frac{\mathbb{E}_{\pi_n} N}{|\Lambda_n|} = \zeta_c(\lambda)$$ for every $\lambda > 0$?
Source: Dickman, Muñoz, Vespignani and Zapperi (2000), for sandpile models, in the form of Hoffman, Johnson and Junge (2024), §1.1; for activated random walk on ℤᵈ also Levine and Silvestri (2024), Conjectures 11 and 12. Stated for the square lattice by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- activated random walk, AIM Open Applied Problems, interacting particle systems, self-organized criticality
References
@article{dickman2000,
author = {Dickman, Ronald and Mu{\~n}oz, Miguel A. and Vespignani, Alessandro and Zapperi, Stefano},
title = {Paths to self-organized criticality},
journal = {Braz. J. Phys.},
volume = {30},
number = {1},
year = {2000},
pages = {27--41},
doi = {10.1590/S0103-97332000000100004},
}
@misc{hoffman2024,
author = {Hoffman, Christopher and Johnson, Tobias and Junge, Matthew},
title = {The density conjecture for activated random walk},
year = {2024},
eprint = {2406.01731},
archivePrefix = {arXiv},
}
@article{levine2024,
author = {Levine, Lionel and Silvestri, Vittoria},
title = {Universality conjectures for activated random walk},
journal = {Probab. Surv.},
volume = {21},
year = {2024},
pages = {1--27},
doi = {10.1214/24-PS25},
eprint = {2306.01698},
archivePrefix = {arXiv},
}
@article{rolla2019,
author = {Rolla, Leonardo T. and Sidoravicius, Vladas and Zindy, Olivier},
title = {Universality and sharpness in activated random walks},
journal = {Ann. Henri Poincar{\'e}},
volume = {20},
number = {6},
year = {2019},
pages = {1823--1835},
doi = {10.1007/s00023-019-00797-0},
eprint = {1707.06081},
archivePrefix = {arXiv},
}
@misc{forien2025,
author = {Forien, Nicolas},
title = {A new proof of superadditivity and of the density conjecture for {A}ctivated {R}andom {W}alks on the line},
year = {2025},
eprint = {2502.02579},
archivePrefix = {arXiv},
}
@misc{aim-activated-random-walk-density,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The density conjecture for activated random walk on the square lattice}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/180-activated-random-walk-density.md},
note = {Entry 180 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00405, "The density conjecture for activated random walk on the square lattice", clopen.solutions.