Two stationary phases in a positive-rate nonlinear voter perturbation
For $L \ge 1$ let $N_L = ([-L, L]^3 \cap \mathbb{Z}^3) \setminus \{0\}$, and let $(a_0, a_1, a_2, a_3, a_4) = (0, 2, 3, 11/2, 7)$. For $\varepsilon > 0$ consider the spin system $\eta \in \{0, 1\}^{\mathbb{Z}^3}$ in which each site $x$, at rate $\varepsilon^{-2}$, copies the state of a uniformly chosen nearest neighbour and, independently, flips at rate $$c_L(x, \eta) = \mathbb{E}\bigl[a_{K(x, \eta, Y)}\bigr] + \frac{1}{100},$$ where $Y = (Y_1, \dots, Y_4)$ is sampled uniformly without replacement from $N_L$ and $K(x, \eta, Y) = \#\{j : \eta(x + Y_j) \ne \eta(x)\}$.
Without the constant $1/100$ this is a voter model perturbation, in the sense of Cox, Durrett and Perkins, by a continuous-time, long-range version of the nonlinear voter model of Molofsky, Durrett, Dushoff, Griffeath and Levin, for which the all-$0$ and all-$1$ configurations are absorbing. With $b_1 = 4a_1 - a_4 = 1$ and $b_2 = 6a_2 - 4a_3 = -4$, the reaction term $$\phi(u) = b_1 u(1-u)^4 + b_2 u^2(1-u)^3 - b_2 u^3(1-u)^2 - b_1 u^4(1-u)$$ of the independent-neighbour approximation has unstable zeros at $0$, $1/2$ and $1$ and stable zeros at some $u_* < 1/2$ and $1 - u_*$. For parameters of this kind Huang and Durrett proved that a further rescaling of the model with absorbing states converges to motion by mean curvature in dimensions $d \ge 3$, and conjectured that it has two translation-invariant stationary distributions with densities close to $u_*$ and $1 - u_*$. Durrett and Huang proposed removing the absorbing states by adding a spontaneous flip rate $\alpha$, which changes the reaction term to $\phi(u) + \alpha(1 - 2u)$; for these rates $1/2$ stays unstable while $\alpha < 1/16$.
Problem. Do there exist a finite $L_0$ and, for every $L \ge L_0$, a number $\varepsilon_0(L) > 0$ such that, for $0 < \varepsilon < \varepsilon_0(L)$, the process has two distinct translation-invariant stationary distributions $\nu^-$ and $\nu^+$ with $$\nu^-(\eta(0) = 1) < 1/2 < \nu^+(\eta(0) = 1)?$$
Source: Durrett, Interacting Particle Systems: Ideas, Techniques, Applications (book manuscript, version of 2 September 2026), Open Problem 7.8.1, after Huang and Durrett (2021), Conjecture 1.8, for the model with absorbing states. Stated with explicit rates by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, interacting particle systems, nonlinear voter models
References
@unpublished{durrett2026,
author = {Durrett, Rick},
title = {Interacting Particle Systems: Ideas, Techniques, Applications},
note = {Book manuscript, version of 2 September 2026, Section 7.8},
year = {2026},
url = {https://sites.math.duke.edu/~rtd/PASTA/PASTA0902.pdf},
}
@article{huang2021,
author = {Huang, Xiangying and Durrett, Rick},
title = {Motion by mean curvature in interacting particle systems},
journal = {Probab. Theory Related Fields},
volume = {181},
number = {1-3},
year = {2021},
pages = {489--532},
doi = {10.1007/s00440-021-01082-0},
eprint = {2006.03090},
archivePrefix = {arXiv},
}
@article{cox2013,
author = {Cox, J. Theodore and Durrett, Richard and Perkins, Edwin A.},
title = {Voter model perturbations and reaction diffusion equations},
journal = {Ast{\'e}risque},
number = {349},
year = {2013},
pages = {vi+113},
doi = {10.24033/ast.921},
eprint = {1103.1676},
archivePrefix = {arXiv},
}
@article{molofsky1999,
author = {Molofsky, Jane and Durrett, Richard and Dushoff, Jonathan and Griffeath, David and Levin, Simon},
title = {Local frequency dependence and global coexistence},
journal = {Theor. Popul. Biol.},
volume = {55},
number = {3},
year = {1999},
pages = {270--282},
doi = {10.1006/tpbi.1998.1404},
}
@misc{aim-positive-rate-nonlinear-voter,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Two stationary phases in a positive-rate nonlinear voter perturbation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/186-positive-rate-nonlinear-voter.md},
note = {Entry 186 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-00462, "Two stationary phases in a positive-rate nonlinear voter perturbation", clopen.solutions.