we-solve (beta)

Math-2924 · mathematics · Probability

Cardy crossing probabilities for Poisson–Voronoi percolation

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For critical planar Poisson–Voronoi percolation and every bounded Jordan quadrilateral, do annealed crossing probabilities converge at vanishing mesh to $integral_0^x$ $[u(1-u)]^{-2/3}du$ divided by $integral_0^1$ $[u(1-u)]^{-2/3}du$, where x is the conformal cross ratio of the marked sides?

Short English statement

For critical planar Poisson–Voronoi percolation and every bounded Jordan quadrilateral, do annealed crossing probabilities converge at vanishing mesh to $integral_0^x$ $[u(1-u)]^{-2/3}du$ divided by $integral_0^1$ $[u(1-u)]^{-2/3}du$, where x is the conformal cross ratio of the marked sides?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2924-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2924

Annealed critical Poisson--Voronoi crossing probabilities are claimed to obey Cardy conformal formula in every bounded Jordan quadrilateral. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Cardy crossing probabilities for Poisson–Voronoi percolation

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Annealed critical Poisson--Voronoi crossing probabilities are claimed to obey Cardy conformal formula in every bounded Jordan quadrilateral.

Mathematical statement of the claimed result

Result 1.
For full-plane Poisson--Voronoi cells at intensity $\varepsilon^{-2}$, independently colour each cell black with probability $1/2$. In a bounded Jordan domain $D$, let $a,b,c,d$ be distinct counterclockwise boundary marks; the crossing event requires a connected black subset of $\overline D$ meeting the closed arcs $ab$ and $cd$. Let $x\in(0,1)$ be the image of $b$ under the conformal map taking $(a,c,d)$ to $(0,1,\infty)$ in $\mathbb H$. Averaging both cell geometry and colours, the claim is \[ \lim_{\varepsilon\downarrow0}\mathbb P(\mathcal C_\varepsilon(D;a,b,c,d)) =\frac{\int_0^x[u(1-u)]^{-2/3}\,du}{\int_0^1[u(1-u)]^{-2/3}\,du} =\frac{\Gamma(2/3)x^{1/3}}{\Gamma(1/3)\Gamma(4/3)} {}_2F_1\left(\frac13,\frac23;\frac43;x\right). \] This is the annealed scalar crossing statement, without extra boundary regularity beyond the Jordan condition.

Outline of the argument

Cluster-boundary circuits are decomposed into fully labeled dual components, including isolated edges, to obtain exact orthogonality. Local plugs open closed field sums, boundary potentials and a finite pin menu control transport along actual arcs, and these identities determine the crossing observable.

Formal verification

Family 224 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

Go to contribution →