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Critical and quenched near-critical universality for Poisson–Voronoi percolation

Cardy's formula holds for Poisson–Voronoi percolation, and near-critical thresholds match the triangular lattice.

3 papers September 23, 2026 Explained by AI

Scatter random points in the plane, give each its own cell of nearest territory, and colour each cell black or white by a fair coin. Cardy's formula predicts the chance that black cells cross a shape between two arcs. These three papers claim to prove that formula for this random-cell model. They also claim that, as the colour bias moves near one half, crossing thresholds behave as on the triangular lattice. No computer proof checker has checked them.

First step
1992
Cardy, the conformal crossing formula
Significance
Major
If they hold, they answer the crossing part of Schramm's Voronoi problem and add quenched near-critical universality.
Trust
Claimed, not checked
Field
Probability and statistical mechanics
3 papers

The explanation

Scatter random points over a square, about ten thousand per unit of area. Give every point the patch of plane closer to it than to any other point. These patches are Voronoi cells. Now flip a fair coin for each cell, colouring it black or white. Can you walk from the left side to the right side on black cells only? For a square, symmetry says the chance tends to one half.

Cardy's formula gives the answer for any shape. Mark four points a, b, c, d on its boundary. Map the shape onto the upper half-plane, sending a, c, d to 0, 1 and infinity. The point b lands at some x between 0 and 1. The crossing chance from arc ab to arc cd then tends to F(x), a fixed integral. For x = 1/2, F is exactly 1/2. For x = 0.25, F is about 0.374. For x = 0.1, F is about 0.267.

Smirnov proved the formula for site percolation on the triangular lattice. Schramm's Problem 2.12 asked for the same conformal invariance for Voronoi percolation. Bollobás and Riordan had shown that one half is the critical colour chance for this model. Tassion had given crossing estimates at every scale.

The third paper claims Cardy's formula for Poisson–Voronoi percolation, in every bounded Jordan domain. A Jordan domain is a region bounded by one simple closed curve. The result is annealed: the chance is averaged over both the points and the colours. It does not claim the full interface curves converge. For a fixed rectangle, known concentration results also give a version with the points held fixed.

The other two papers move slightly off the critical point. Picture raising the black chance from one half step by step. Each crossing first appears at some threshold. The window where this happens shrinks as the cells shrink. A pivotal cell is one whose colour decides whether a crossing happens. The second paper, taking Cardy's formula as input, counts these cells. The expected number for a unit-square crossing grows like a constant times ε to the −3/4, where the point density is ε to the −2. With ten thousand points per unit area, ε is 0.01 and ε to the −3/4 is about 31.6.

The first paper uses that count to set the scale. It fixes one random tessellation and averages only the colours. That is called quenched. It claims the joint crossing thresholds for rational polygonal shapes tend to the same law as on the triangular lattice. The obstacle is that a fixed random geometry could, in principle, bend the answer. The paper shows that a typical geometry does not.

These two near-critical results rest on the Cardy formula as an input. So they inherit any gap in it.

How sure is this? No Lean formalization is listed for these papers in the openai/math release. So no computer proof checker has checked them. No specialist review has been published.

What it doesn't show. This does not prove convergence of the full interface curves, or a quenched Cardy formula in every Jordan domain.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if all three papers hold. They build on each other, no Lean formalization is listed, and no specialist review is published.

Words used

Voronoi cell
The patch of plane that is closer to one given point than to any other point.
Percolation
Colouring sites or cells at random and asking whether connected paths of one colour cross a region.
Cardy's formula
An exact formula for the chance that a critical percolation cluster crosses between two arcs of a shape.
Annealed
Averaged over all the randomness at once, here both the points and the colours.
Quenched
With the random geometry fixed first, averaging only over the colours.
Pivotal cell
A cell whose colour decides whether a given crossing happens.
Near-critical
Slightly away from the critical colour chance, on the scale where large crossings start to change.

The original papers (3)

All quoted lines (10)
  1. For v∈ηv\in\eta its Voronoi cell is the closed set Vv={z∈C:∣z−v∣≤∣z−u∣ for every u∈η}.\mathcal V_v=\{z\in\C:|z-v|\le |z-u|\text{ for every }u\in\eta\}.
    Cardy’s formula for critical Poisson–Voronoi percolation Back to text
  2. Its value at bb is a uniquely determined number x∈(0,1)x\in(0,1).
    Cardy’s formula for critical Poisson–Voronoi percolation Back to text
  3. Smirnov proved the formula for site percolation on the triangular lattice.
    Cardy’s formula for critical Poisson–Voronoi percolation Back to text
  4. Schramm's Problem 2.12 asks for the corresponding conformal-invariance theorem for Voronoi percolation.
    Cardy’s formula for critical Poisson–Voronoi percolation Back to text
  5. We prove the annealed Cardy-crossing aspect of this conjecture for the Poisson model, with no regularity assumption on the boundary beyond the Jordan property.
    Cardy’s formula for critical Poisson–Voronoi percolation Back to text
  6. The statement does not assert convergence of the full interface law or a quenched crossing limit for arbitrary Jordan domains.
    Cardy’s formula for critical Poisson–Voronoi percolation Back to text
  7. For a fixed rectangle, the established concentration of conditional crossing probabilities also gives a quenched L2L^2 limit; see Corollary … below.
    Cardy’s formula for critical Poisson–Voronoi percolation Back to text
  8. 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\eps^{-3/4}, where the point intensity is ε−2\eps^{-2}.
    A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula Back to text
  9. Conditional on the Poisson tessellation, the threshold law converges in probability over tessellations to the same law as on the triangular lattice.
    From critical crossings to quenched near-critical universality in Voronoi percolation Back to text
  10. We prove such a transfer for Poisson–Voronoi percolation, taking its scalar critical Cardy formula as an explicit input.
    From critical crossings to quenched near-critical universality in Voronoi percolation Back to text

Other sources

  1. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model
  2. J. L. Cardy, “Critical percolation in finite geometries”, Journal of Physics A 25 (1992), L201–L206
  3. S. Smirnov, “Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits”, Comptes Rendus de l'Académie des Sciences 333 (2001), 239–244

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