Cardy's crossing formula for square-lattice bond percolation
Let $D \subset \mathbb{C}$ be a bounded simply connected domain with smooth boundary and four distinct boundary points $a, b, c, d$ in counterclockwise order. For $\delta > 0$, declare each edge of the induced nearest-neighbour graph on $D \cap \delta\mathbb{Z}^2$ open independently with probability $1/2$, and let $P_\delta$ be the probability that an open path joins a vertex within distance $2\delta$ of the boundary arc $ab$ to a vertex within distance $2\delta$ of the boundary arc $cd$. Let $\varphi$ be the conformal map of $D$ onto the upper half-plane with $\varphi(a) = 0$, $\varphi(c) = 1$ and $\varphi(d) = \infty$, and put $\eta = \varphi(b) \in (0, 1)$.
Cardy predicted that the limiting crossing probabilities of critical percolation depend only on the conformal type of the marked domain, and gave them explicitly in terms of hypergeometric functions; Smirnov proved this for critical site percolation on the triangular lattice. For bond percolation on $\mathbb{Z}^2$ at $p = 1/2$, Duminil-Copin, Kozlowski, Krachun, Manolescu and Oulamara proved that crossing probabilities of rectangles are asymptotically invariant under rotations.
Problem. Is it true that $$\lim_{\delta \downarrow 0} P_\delta = \frac{\Gamma(2/3)}{\Gamma(1/3)\,\Gamma(4/3)}\, \eta^{1/3}\, {}_2F_1\Bigl(\frac13, \frac23; \frac43; \eta\Bigr)?$$
Source: Tassion (2023), Conjecture 3.1, as convergence of crossing probabilities of conformal rectangles to a conformally invariant limit, whose explicit form is Cardy's (1992). Stated by AIM — Open Applied Problems for smooth domains, with this discretization.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, conformal invariance, critical percolation, crossing probabilities
References
@article{tassion2024,
author = {Tassion, Vincent},
title = {Expos{\'e} {B}ourbaki 1210: {R}otation invariance of critical planar percolation [after {H}ugo {D}uminil-{C}opin, {K}arol {K}ajetan {K}ozlowski, {D}mitry {K}rachun, {I}oan {M}anolescu and {M}endes {O}ulamara]},
journal = {Ast{\'e}risque},
volume = {446},
pages = {481--499},
year = {2024},
doi = {10.24033/ast.1219},
note = {S{\'e}minaire Bourbaki, 75e ann{\'e}e, 2022--2023, no.~1210, June 2023},
url = {https://www.bourbaki.fr/TEXTES/Exp1210-Tassion.pdf},
}
@article{cardy1992,
author = {Cardy, John L.},
title = {Critical percolation in finite geometries},
journal = {J. Phys. A},
volume = {25},
number = {4},
year = {1992},
pages = {L201--L206},
doi = {10.1088/0305-4470/25/4/009},
eprint = {hep-th/9111026},
archivePrefix = {arXiv},
}
@article{smirnov2001,
author = {Smirnov, Stanislav},
title = {Critical percolation in the plane: conformal invariance, {C}ardy's formula, scaling limits},
journal = {C. R. Acad. Sci. Paris S{\'e}r. I Math.},
volume = {333},
number = {3},
year = {2001},
pages = {239--244},
doi = {10.1016/S0764-4442(01)01991-7},
}
@misc{duminilcopin2020rotation,
author = {Duminil-Copin, Hugo and Kozlowski, Karol Kajetan and Krachun, Dmitry and Manolescu, Ioan and Oulamara, Mendes},
title = {Rotational invariance in critical planar lattice models},
year = {2020},
eprint = {2012.11672},
archivePrefix = {arXiv},
}
@misc{aim-square-bond-cardy-formula,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Cardy's crossing formula for square-lattice bond percolation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/090-square-bond-cardy-formula.md},
note = {Entry 090 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
For every fixed $1\le q<4$, critical square-lattice random-cluster Dobrushin interfaces converge as ordered curves to chordal $\SLE_{\kappa(q)}$, where $\kappa(q)=4\pi/\arccos(-\sqrt q/2)$, confirming the Rohde--Schramm prediction in this parameter range. This holds in every bounded Jordan domain under uniform marked boundary approximation. The complete nested plane loop collections converge to whole-plane $\CLE_{\kappa(q)}$ in a spherical matching topology retaining multiplicities and traversals. At …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00719, "Cardy's crossing formula for square-lattice bond percolation", clopen.solutions.