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From critical crossings to quenched near-critical universality in Voronoi percolation
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Abstract
Taking Cardy's crossing formula for critical Poisson–Voronoi percolation as an input, we prove a universal joint limit for the near-critical crossing thresholds of rational polygonal quadrilaterals under the monotone coupling. Conditional on the Poisson tessellation, the threshold law converges in probability over tessellations to the same law as on the triangular lattice. Each model is normalized by its own expected number of color-pivotal sites for a unit-square crossing.
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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). Treats pivotal processes and annealed spectral samples for Poisson–Voronoi crossing events; possibly relevant to the paper's pivotal-count normalization and its comparison with Vanneuville's annealed spectral sample (Section 1).
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:From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026,
author = {{OpenAI}},
title = {{From critical crossings to quenched near-critical universality in Voronoi percolation}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026/critical-crossings-quenched-near-critical-universality-voronoi-percolation.pdf}{OAI:From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-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:
- A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula · October 5, 2026
- Cardy’s formula for critical Poisson–Voronoi percolation · September 23, 2026