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Cardy’s formula for critical Poisson–Voronoi percolation
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Abstract
We prove Cardy's formula for annealed crossing probabilities in critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral. This proves the annealed crossing-probability form of the conformal-invariance conjecture for this model.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Possibly relevant literature (found by the agent, not checked)
- F. Camia, C. M. Newman, V. Sidoravicius, Cardy's Formula for some Dependent Percolation Models (arXiv:cond-mat/0111293, 2001). An earlier extension of Cardy's formula beyond independent triangular-lattice site percolation; relevant to the paper's claim (Section 1.2) of carrying a Smirnov-type crossing argument to a different model.
- M. Khristoforov, M. Skopenkov, S. Smirnov, A generalization of Cardy's and Schramm's formulae (arXiv:2305.12368; Commun. Math. Phys. 406, 105 (2025)). Constructs a new discrete analytic observable for triangular-lattice percolation; related to the paper's observable-based identification of the limit as a boundary-value problem (Section 8).
- D. Ahlberg, D. de la Riva, S. Griffiths, On the rate of convergence in quenched Voronoi percolation (arXiv:2103.01870, 2021). Gives concentration rates for conditional crossing probabilities given the tessellation; related to the quenched $L^2$ limit for rectangles (Corollary 1.2), which the paper derives from Vanneuville's variance bound.
Digests and expositions
- Latent Seminar, Voronoi percolation: crossings, pivotals, and near-critical laws (expository notes, 2026-10-06). Explains the three manuscripts' claimed results, their conditional dependencies and selected proof mechanisms, explicitly without independently verifying the complete proofs. [family-level]
- heyprof, Cardy's formula — percolation in a random tiling (interactive exposition, undated). Introduces the manuscript's claimed annealed crossing limit and describes a simulation comparing Voronoi crossing frequencies with Cardy's prediction.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 224
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Cardys-formula-for-critical-Poisson-Voronoi-percolation-September-23-2026,
author = {{OpenAI}},
title = {{Cardy's formula for critical Poisson--Voronoi percolation}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Cardys-formula-for-critical-Poisson-Voronoi-percolation-September-23-2026/paper.pdf}{OAI:Cardys-formula-for-critical-Poisson-Voronoi-percolation-September-23-2026}},
year = {2026}
}Family 224: Critical and quenched near-critical universality for Poisson–Voronoi percolation
Proves Cardy's formula for annealed critical Poisson–Voronoi crossing probabilities in every bounded Jordan quadrilateral. With each model normalized by its own expected unit-square pivotal count, the conditional joint near-critical crossing-threshold laws for rational polygonal quads converge in environment probability to the triangular-lattice reference law. This establishes quenched near-critical universality for crossing thresholds.
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