Result 224, Probability and statistical mechanics

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.

Proof

The bigger picture

Why it matters

In Poisson-Voronoi percolation, random points divide the plane into nearest-point cells, which are randomly colored. The manuscripts report that large-scale crossing behavior matches a standard lattice model, despite the irregular geometry.

What changes?

The manuscripts report Cardy's shape-dependent formula for critical crossing probabilities in every bounded Jordan quadrilateral: a region with a simple closed boundary and four marked corners. Probabilities average over tessellations and colors. For polygonal quadrilaterals with rational vertices, joint near-critical crossing thresholds, measured as one color is added monotonically, reportedly converge to the triangular-lattice reference law. This convergence holds for conditional laws in probability over tessellations, with each model normalized by its own expected unit-square pivotal count.

What does that help mathematicians do?

A pivotal cell is one whose color change switches whether a crossing exists. Taking Cardy's formula as input, the companion manuscript reports that the expected unit-square pivotal count is asymptotic to a positive constant times epsilon to the power minus three quarters, at point intensity epsilon to the power minus two. This supplies a precise normalization scale without requiring a convergence rate in Cardy's formula, connecting critical geometry to sensitivity under changes in coloring probability.

Are there practical applications?

The immediate value is foundational for probability and statistical mechanics: the claimed conditional limit shows how crossing-threshold statistics in a fixed random geometry approach a common reference law, in probability over those geometries. This goes beyond agreement after averaging over environments. Its scope is crossing thresholds, not universality for every observable or a demonstrated practical technology.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

From critical crossings to quenched near-critical universality in Voronoi percolation

October 5, 2026 61 pages

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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Cite (BibTeX) 
@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}
}

A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula

October 5, 2026 43 pages

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 ε−3/4\varepsilon ^{-3/4}, where the point intensity is ε−2\varepsilon ^{-2}. No rate of convergence in Cardy's formula is required.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Cardy’s formula for critical Poisson–Voronoi percolation

September 23, 2026 75 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.