A phase transition for continuum hard disks
Let $\Omega$ be the set of locally finite sets $\omega \subset \mathbb{R}^2$ with $|x - y| \ge 1$ for all distinct $x, y \in \omega$. For an activity $z > 0$, a hard-disk Gibbs law is a probability measure $\mu$ on $\Omega$ such that, for every bounded Borel set $\Lambda$, its conditional law inside $\Lambda$ given the configuration outside is the Poisson process of intensity $z$ on $\Lambda$, conditioned on the combined configuration having all pair distances at least one.
Michelen and Perkins proved that the hard-disk Gibbs law is unique, and the pressure analytic, for every activity below an explicit constant larger than $e/\pi$. Richthammer proved that at every activity all Gibbs laws of the hard-disk model are translation invariant. In simulations, Alder and Wainwright observed a phase transition, and Bernard and Krauth a first-order liquid–hexatic transition.
Problem. Does there exist a finite activity $z > 0$ for which two distinct infinite-volume hard-disk Gibbs laws exist? The particles have only the hard-core interaction: no extra attraction, lattice restriction or particle labels are allowed.
Source: Stated by AIM — Open Applied Problems, after Michelen and Perkins (2025), §1 and §1.1.1; a phase transition for hard disks was first reported in the simulations of Alder and Wainwright (1962).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Gibbs point processes, hard disks, phase transitions
References
@article{michelen2025,
author = {Michelen, Marcus and Perkins, Will},
title = {Potential-weighted connective constants and uniqueness of {G}ibbs measures},
journal = {Comm. Math. Phys.},
volume = {406},
number = {2},
year = {2025},
pages = {Paper No. 32},
doi = {10.1007/s00220-024-05214-0},
eprint = {2109.01094},
archivePrefix = {arXiv},
}
@article{alder1962,
author = {Alder, B. J. and Wainwright, T. E.},
title = {Phase transition in elastic disks},
journal = {Phys. Rev.},
volume = {127},
number = {2},
year = {1962},
pages = {359--361},
doi = {10.1103/PhysRev.127.359},
}
@article{richthammer2007,
author = {Richthammer, Thomas},
title = {Translation-invariance of two-dimensional {G}ibbsian point processes},
journal = {Comm. Math. Phys.},
volume = {274},
number = {1},
year = {2007},
pages = {81--122},
doi = {10.1007/s00220-007-0274-7},
eprint = {0706.3637},
archivePrefix = {arXiv},
}
@article{bernard2011,
author = {Bernard, Etienne P. and Krauth, Werner},
title = {Two-step melting in two dimensions: first-order liquid-hexatic transition},
journal = {Phys. Rev. Lett.},
volume = {107},
number = {15},
year = {2011},
pages = {155704},
doi = {10.1103/PhysRevLett.107.155704},
}
@misc{aim-hard-disk-phase-transition,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A phase transition for continuum hard disks}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/187-hard-disk-phase-transition.md},
note = {Entry 187 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-00439, "A phase transition for continuum hard disks", clopen.solutions.