Consensus in the one-dimensional Axelrod model above the feature threshold
Fix integers $F > q \ge 3$. In the Axelrod model on $\mathbb{Z}$, each site $x$ carries a culture $\eta_t(x) \in \{1, \dots, q\}^F$, a vector of $F$ features. Initially all features at all sites are independent and uniform on $\{1, \dots, q\}$. Each nearest-neighbour edge $\{x, y\}$ rings at rate one; if $\eta_t(x)$ and $\eta_t(y)$ agree on $a$ features with $0 < a < F$, then with probability $a/F$ a feature on which they disagree is chosen uniformly at random, one of the two sites is chosen uniformly at random, and that site copies the other's value of the feature; otherwise nothing happens. The model is due to Axelrod. Lanchier proved that the model clusters, in the sense below, when $F = q = 2$, and Lanchier and Schweinsberg when $q = 2$ and $F \ge 2$ is arbitrary. Lanchier and Scarlatos proved that it fixates, and does not cluster, when $F \le cq$ with $c \approx 0.567$ the solution of $e^{-c} = c$, and when $F = 2$ and $q \ge 3$.
Problem. Is it true that $$\lim_{t \to \infty} \mathbb{P}\bigl(\eta_t(x) \ne \eta_t(y)\bigr) = 0$$ for every fixed pair of sites $x, y \in \mathbb{Z}$? This concerns the convergence of these probabilities, not almost sure stabilisation of each site.
Source: Lanchier and Schweinsberg (2012), Introduction, who conjecture clustering for all F > q from simulations of Vilone, Vespignani and Castellano (2002) and prove the case q = 2; AIM — Open Applied Problems states the remaining cases q ≥ 3.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Keywords
- AIM Open Applied Problems, Axelrod model, clustering, interacting particle systems
References
@article{lanchier2012consensus,
author = {Lanchier, Nicolas and Schweinsberg, Jason},
title = {Consensus in the two-state {A}xelrod model},
journal = {Stochastic Process. Appl.},
volume = {122},
number = {11},
year = {2012},
pages = {3701--3717},
doi = {10.1016/j.spa.2012.06.010},
eprint = {1107.4413},
archivePrefix = {arXiv},
}
@article{vilone2002,
author = {Vilone, D. and Vespignani, A. and Castellano, C.},
title = {Ordering phase transition in the one-dimensional {A}xelrod model},
journal = {Eur. Phys. J. B},
volume = {30},
number = {3},
year = {2002},
pages = {399--406},
doi = {10.1140/epjb/e2002-00395-2},
}
@article{axelrod1997,
author = {Axelrod, Robert},
title = {The dissemination of culture: a model with local convergence and global polarization},
journal = {J. Conflict Resolut.},
volume = {41},
number = {2},
year = {1997},
pages = {203--226},
doi = {10.1177/0022002797041002001},
}
@article{lanchier2012revisited,
author = {Lanchier, Nicolas},
title = {The {A}xelrod model for the dissemination of culture revisited},
journal = {Ann. Appl. Probab.},
volume = {22},
number = {2},
year = {2012},
pages = {860--880},
doi = {10.1214/11-AAP790},
eprint = {1004.0365},
archivePrefix = {arXiv},
}
@article{lanchier2013,
author = {Lanchier, Nicolas and Scarlatos, Stylianos},
title = {Fixation in the one-dimensional {A}xelrod model},
journal = {Ann. Appl. Probab.},
volume = {23},
number = {6},
year = {2013},
pages = {2538--2559},
doi = {10.1214/12-AAP910},
eprint = {1301.0130},
archivePrefix = {arXiv},
}
@misc{aim-axelrod-multifeature-clustering,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Consensus in the one-dimensional Axelrod model above the feature threshold}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/283-axelrod-multifeature-clustering.md},
note = {Entry 283 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-00407, "Consensus in the one-dimensional Axelrod model above the feature threshold", clopen.solutions.