A deterministic nontrivial growth exponent for planar DLA
Let $A_1 = \{0\} \subset \mathbb{Z}^2$ and, given $A_n$, let $A_{n+1} = A_n \cup \{y\}$, where $y$ is chosen from the exterior vertex boundary of $A_n$ (the sites outside $A_n$ adjacent to it) according to harmonic measure from infinity: the limit of the first-hitting distribution of that boundary for simple random walk started far away. This is external diffusion-limited aggregation (DLA), introduced by Witten and Sander. Let $R_n = \max\{|x| : x \in A_n\}$, with $|\cdot|$ the Euclidean norm. Since $A_n$ has $n$ sites, $R_n \ge n^{1/2}/3$ for $n \ge 2$. Kesten proved that almost surely $\limsup_{n \to \infty} n^{-2/3} R_n < \infty$, and Losev and Smirnov gave another proof of this bound and proved analogous bounds, depending on the model's parameter, for the dielectric-breakdown model.
Problem. Is there a deterministic $\beta \in (1/2, 2/3]$ such that $$\lim_{n \to \infty} \frac{\log R_n}{\log n} = \beta \qquad \text{almost surely?}$$
Source: Stated by AIM — Open Applied Problems, after the conjecture recorded by Losev and Smirnov (2025), §1.1, that the growth rate of diffusion-limited aggregation exists almost surely, is nontrivial and does not depend on the realization; the model is due to Witten and Sander (1981).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, diffusion-limited aggregation, harmonic measure, random growth models
References
@article{losev2025,
author = {Losev, Ilya and Smirnov, Stanislav},
title = {How long are the arms in {DBM}?},
journal = {Comm. Math. Phys.},
volume = {406},
number = {4},
year = {2025},
pages = {93},
doi = {10.1007/s00220-025-05276-8},
eprint = {2307.14931},
archivePrefix = {arXiv},
}
@article{witten1981,
author = {Witten, T. A. and Sander, L. M.},
title = {Diffusion-limited aggregation, a kinetic critical phenomenon},
journal = {Phys. Rev. Lett.},
volume = {47},
number = {19},
year = {1981},
pages = {1400--1403},
doi = {10.1103/PhysRevLett.47.1400},
}
@article{kesten1987,
author = {Kesten, Harry},
title = {How long are the arms in {DLA}?},
journal = {J. Phys. A},
volume = {20},
number = {1},
year = {1987},
pages = {L29--L33},
doi = {10.1088/0305-4470/20/1/007},
}
@misc{aim-dla-growth-exponent,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A deterministic nontrivial growth exponent for planar DLA}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/177-dla-growth-exponent.md},
note = {Entry 177 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-00426, "A deterministic nontrivial growth exponent for planar DLA", clopen.solutions.