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

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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.

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The paper states

Theorem 1.2 (Quenched near-critical universality)

In From critical crossings to quenched near-critical universality in Voronoi percolation, 5 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Three other possible matches are listed below the card, among them Theorem 1.2 in A Pivotal Amplitude for Voronoi Percolation from Cardy’s Formula and Corollary 1.2 (Quenched Cardy limit for fixed rectangles) in Cardy’s formula for critical Poisson–Voronoi percolation.

Assume Hypothesis 1.1. The expectations in (1) are finite and strictly positive for every mesh, and both scales tend to zero as their mesh tends to zero. There is a probability measure \nu_\triangle on \mathcal K such that

\Law(\tau_\delta^\triangle)\Longrightarrow\nu_\triangle \qquad(\delta\downarrow0).

For the independently normalized Voronoi model,

d_{\mathrm{BL}}\bigl(\Law(\tau_\varepsilon^V\mid\eta_\varepsilon), \nu_\triangle\bigr) \xrightarrow[\varepsilon\downarrow0]{\PP}0.

The conditional law averages all color marks and no geometry.

Three more possible matches

  • Theorem 1.2 in A Pivotal Amplitude for Voronoi Percolation from Cardy’s Formula, 5 October 2026.
  • Corollary 1.2 (Quenched Cardy limit for fixed rectangles) in Cardy’s formula for critical Poisson–Voronoi percolation, 23 September 2026.
  • Theorem 1.1 (Annealed Voronoi crossing law) in Cardy’s formula for critical Poisson–Voronoi percolation, 23 September 2026.
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From critical crossings to quenched near-critical universality in Voronoi percolation

5 October 2026, current edition.

Read it herePDF

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.

A Pivotal Amplitude for Voronoi Percolation from Cardy’s Formula

5 October 2026, current edition.

Read it herePDF

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

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