OP-00673 Solution claimed

Absence of an infinite cluster at criticality in three-dimensional percolation

Consider Bernoulli bond percolation with parameter $p \in [0,1]$ on the nearest-neighbour cubic lattice $\mathbb{Z}^3$. Let $\theta(p)$ be the probability that the open cluster of the origin is infinite, and $p_c = \inf\{p \in [0,1] : \theta(p) > 0\}$. On $\mathbb{Z}^2$, Harris and Kesten proved that $\theta(p_c) = 0$; on $\mathbb{Z}^d$ with $d \ge 11$ it follows from the mean-field behaviour proved by Fitzner and van der Hofstad. On $\mathbb{Z}^3$, Barsky, Grimmett and Newman proved that at $p_c$ there is almost surely no infinite open cluster in the half-space $\mathbb{Z}^2 \times \{0, 1, 2, \dots\}$, and Duminil-Copin, Sidoravicius and Tassion that for every $k$ there is none in the slab $\mathbb{Z}^2 \times \{0, \dots, k\}$ at the slab's own critical point; these critical points converge to $p_c$ as $k \to \infty$ (Grimmett and Marstrand).

Problem. Is $\theta(p_c) = 0$?

Source: Classical, with no single originator; the case d = 3 of the θ(p_c) = 0 conjecture, see Duminil-Copin, Sixty years of percolation (ICM 2018), Conjecture 1.

Imported from github.com/MColbrook/AIM.

Classification

Primary: PR — Probability
Secondary: MP
Keywords: AIM Open Applied Problems, critical phenomena, percolation

References

@incollection{duminilcopin2019,
  author        = {Duminil-Copin, Hugo},
  title         = {Sixty years of percolation},
  booktitle     = {Proceedings of the International Congress of Mathematicians (ICM 2018)},
  publisher     = {World Scientific},
  year          = {2019},
  pages         = {2829--2856},
  doi           = {10.1142/9789813272880_0162},
  eprint        = {1712.04651},
  archivePrefix = {arXiv},
}
@article{harris1960,
  author  = {Harris, T. E.},
  title   = {A lower bound for the critical probability in a certain percolation process},
  journal = {Math. Proc. Cambridge Philos. Soc.},
  volume  = {56},
  number  = {1},
  year    = {1960},
  pages   = {13--20},
  doi     = {10.1017/S0305004100034241},
}
@article{kesten1980,
  author  = {Kesten, Harry},
  title   = {The critical probability of bond percolation on the square lattice equals $\frac12$},
  journal = {Comm. Math. Phys.},
  volume  = {74},
  number  = {1},
  year    = {1980},
  pages   = {41--59},
  doi     = {10.1007/BF01197577},
}
@article{fitzner2017,
  author        = {Fitzner, Robert and van der Hofstad, Remco},
  title         = {Mean-field behavior for nearest-neighbor percolation in $d > 10$},
  journal       = {Electron. J. Probab.},
  volume        = {22},
  year          = {2017},
  pages         = {1--65},
  doi           = {10.1214/17-EJP56},
  eprint        = {1506.07977},
  archivePrefix = {arXiv},
}
@article{barsky1991,
  author  = {Barsky, David J. and Grimmett, Geoffrey R. and Newman, Charles M.},
  title   = {Percolation in half-spaces: equality of critical densities and continuity of the percolation probability},
  journal = {Probab. Theory Related Fields},
  volume  = {90},
  number  = {1},
  year    = {1991},
  pages   = {111--148},
  doi     = {10.1007/BF01321136},
}
@article{duminilcopin2016,
  author        = {Duminil-Copin, Hugo and Sidoravicius, Vladas and Tassion, Vincent},
  title         = {Absence of infinite cluster for critical {B}ernoulli percolation on slabs},
  journal       = {Comm. Pure Appl. Math.},
  volume        = {69},
  number        = {7},
  year          = {2016},
  pages         = {1397--1411},
  doi           = {10.1002/cpa.21641},
  eprint        = {1401.7130},
  archivePrefix = {arXiv},
}
@article{grimmett1990,
  author  = {Grimmett, G. R. and Marstrand, J. M.},
  title   = {The supercritical phase of percolation is well behaved},
  journal = {Proc. Roy. Soc. London Ser. A},
  volume  = {430},
  number  = {1879},
  year    = {1990},
  pages   = {439--457},
  doi     = {10.1098/rspa.1990.0100},
}
@misc{aim-critical-percolation-three-dimensions,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Absence of an infinite cluster at criticality in three-dimensional percolation}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/088-critical-percolation-three-dimensions.md},
  note         = {Entry 088 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Critical bond and site percolation on the cubic lattice v1 GitHub ↗

OpenAI

We prove that nearest-neighbor Bernoulli bond and site percolation on $\mathbb Z^3$ have no infinite cluster at their respective critical parameters. The proof combines a finite connection inequality for independent hyperedges with a finite-scale extension estimate and an adaptive exploration.

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: math @ 1581d4c, build/critical-percolation-z3.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00673, "Absence of an infinite cluster at criticality in three-dimensional percolation", clopen.solutions.