Browse › Probability and statistical mechanics › 224 Critical and quenched near-critical universality for Poisson–Voronoi percolation
A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
Taking Cardy's conformal crossing formula for critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral as an input, we prove that the expected number of color-pivotal cells for a unit-square crossing is asymptotic to a positive constant times $\varepsilon ^{-3/4}$, where the point intensity is $\varepsilon ^{-2}$. No rate of convergence in Cardy's formula is required.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
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)
- C. Bhattacharjee, G. Peccati, D. Yogeshwaran, Spectra of Poisson functionals and applications in continuum percolation (arXiv:2407.13502, 2024). Relates pivotal processes and spectral samples to crossing events in Poisson–Voronoi percolation; possibly relevant to the paper's color-pivotal count and its differentiation of the pivotal probability with respect to Poisson intensity (Section 6).
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]
Elsewhere
- Emergent Mind: family 224
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026,
author = {{OpenAI}},
title = {{A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf}{OAI:A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026}},
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.
Other papers in this family:
- From critical crossings to quenched near-critical universality in Voronoi percolation · October 5, 2026
- Cardy’s formula for critical Poisson–Voronoi percolation · September 23, 2026