1 · The essentials
Imagine a plane divided into randomly shaped cells, each given one of two colors, and ask whether one color connects two specified sides of a region. These manuscripts study that question at a critical color balance and as the balance changes slightly. The third manuscript shows that, at the critical balance, crossing probabilities averaged over both the random cells and their colors follow Cardy’s formula. The main manuscript takes that formula as an input and studies the color balance at which crossings first appear in several regions. After each model is rescaled using its own expected count of cells or sites whose color can change a square-crossing outcome, the manuscript shows that these crossing thresholds approach the same joint law as in the triangular-lattice model. This agreement concerns the color randomness even when the random division into cells is held fixed, with convergence measured over those divisions. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, picture two mosaics—one with irregular pieces and one with a regular layout—whose pieces gradually change color until colored routes connect selected edges. The comparison uses a separate calibration for each mosaic, based on how many individual pieces can decide whether a route exists, rather than assuming that the same raw adjustment means the same thing in both.
Why it matters
The family connects a formula for critical crossing probabilities to a joint description of when crossings appear near criticality. Its main conclusion goes beyond averaging over the random geometry: the conditional threshold law approaches the triangular-lattice reference law in probability over the geometry. The companion pivotal-count result identifies the scale of the normalization used for Voronoi percolation.
2 · Going further
The critical input concerns planar Poisson–Voronoi percolation: a Poisson tessellation supplies the cells, and their colors determine crossings. Manuscript 3 shows Cardy’s formula for annealed critical crossing probabilities in every bounded Jordan quadrilateral. Here, “annealed” means that the probability averages over both the tessellation and the colors. The source describes this as the annealed crossing-probability form of the conformal-invariance conjecture for this model.
The main manuscript takes that critical formula as an input and considers the joint near-critical crossing thresholds of rational polygonal quadrilaterals under a monotone coupling. Such a coupling lets the color parameter change while preserving crossings once they appear. Each model is normalized by its own expected number of color-pivotal cells or sites for a unit-square crossing, where a pivotal color is one whose change alters the crossing outcome. Conditional on the Poisson tessellation, the resulting joint threshold law is shown in the manuscript to converge, in probability over tessellations, to the triangular-lattice reference law. This is the stated quenched near-critical universality conclusion.
Manuscript 2 supplies a related quantitative statement about pivotal counts. Assuming Cardy’s formula in every bounded Jordan quadrilateral, it shows that, for Poisson point intensity , the expected number of color-pivotal cells for a unit-square crossing is asymptotic to for a positive constant . No rate of convergence in Cardy’s formula is required for this statement. Together, the abstracts describe a route from critical crossing information to pivotal normalization and conditional near-critical threshold laws; they do not provide the detailed argument or state that Manuscript 2 is a required intermediate step in the main manuscript.
Useful notions
- Probabilities and conditional probabilities
- Random tessellations and colored-cell crossings
- Critical and near-critical parameters
- Joint distributions and convergence in probability
- Asymptotic scaling
What was known before / what is new
Before
The main manuscript and the pivotal-amplitude manuscript both take Cardy’s formula for critical Poisson–Voronoi crossings as an input. The source identifies the triangular-lattice law as the reference for comparison and describes the critical crossing statement as a form of the conformal-invariance conjecture. It does not give a separate historical account of which results preceded this manuscript family.
New
Manuscript 3 shows the annealed Cardy crossing formula for every bounded Jordan quadrilateral in the stated model. Manuscript 2 shows a positive-constant asymptotic for the expected unit-square pivotal count, without requiring a convergence rate in that formula. The main manuscript shows convergence of conditional joint near-critical crossing-threshold laws to the triangular-lattice reference law, using each model’s own pivotal-count normalization.
Limits and assumptions
The main result assumes the critical Cardy formula and concerns joint crossing thresholds for rational polygonal quadrilaterals under the monotone coupling. Its conditional-law convergence is in probability over Poisson tessellations; the stated result is not an almost-sure convergence claim. The broader bounded-Jordan-quadrilateral scope belongs to the annealed critical formula, not to the stated near-critical threshold conclusion. The pivotal asymptotic also assumes that critical formula and specifies a positive constant without giving its value in the abstract. The main result is not formalized in Lean and has not been formally verified. The supplied text does not identify further open problems or claim a broader form of conformal invariance.
Mini glossary
- Annealed crossing probability —
- A crossing probability averaged over both the random tessellation and the colors.
- Quenched threshold law —
- The distribution of crossing thresholds conditional on a fixed tessellation, leaving the color randomness.
- Color-pivotal cell —
- A cell whose color change changes whether the specified crossing occurs.
- Monotone coupling —
- A shared construction across color-parameter values in which an existing crossing is preserved as the parameter increases.
- Joint crossing-threshold law —
- The distribution of the parameter values at which crossings first appear in several regions, considered together.
Keywords : Poisson–Voronoi percolation · Cardy’s formula · Quenched near-critical universality
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
From critical crossings to quenched near-critical universality in Voronoi percolation
05/10/2026Taking 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.
A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula
05/10/2026Taking 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 , where the point intensity is . No rate of convergence in Cardy's formula is required.
Cardy’s formula for critical Poisson–Voronoi percolation
23/09/2026We 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.