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

What OpenAI claims

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.

Papers (3)

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

   5 October 2026 · Preprint, not peer reviewed · PDF

   OpenAI's 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.

2. A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula

   5 October 2026 · Preprint, not peer reviewed · PDF

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

3. Cardy’s formula for critical Poisson–Voronoi percolation

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.