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Math-2926 · mathematics · Probability

The Voronoi pivotal-count amplitude under Cardy’s formula

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Assuming the stated full-domain Cardy crossing limit for critical Poisson–Voronoi percolation, does $\epsilon ^{3/4}$ E $N_\epsilon $ tend to a finite positive constant for the source-defined reference quad pivotal count?

Short English statement

Assuming the stated full-domain Cardy crossing limit for critical Poisson–Voronoi percolation, does $\epsilon ^{3/4}$ E $N_\epsilon $ tend to a finite positive constant for the source-defined reference quad pivotal count?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2926-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2926

Assuming the Voronoi Cardy input without a convergence rate, the expected number of color pivotals is claimed to be asymptotic to $C\varepsilon^{-3/4}$ with $C>0$. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The Voronoi pivotal-count amplitude under Cardy’s formula

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Assuming the Voronoi Cardy input without a convergence rate, the expected number of color pivotals is claimed to be asymptotic to $C\varepsilon^{-3/4}$ with $C>0$.

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 $\mathbb H$. Assume these scalar crossing probabilities have Cardy's limit for every bounded Jordan quadrilateral. Let $N_\varepsilon$ count nuclei whose colour flip changes the left-to-right crossing of $[0,1]^2$, including outside nuclei whose cells meet that square. There exists $c_V\in(0,\infty)$ with \[ \varepsilon^{3/4}\mathbb EN_\varepsilon\longrightarrow c_V,\qquad (\mathbb EN_\varepsilon)^{-1}\sim c_V^{-1}\varepsilon^{3/4}. \] Expectation averages geometry and colours. The hypothesis does not assume an interface limit or a convergence rate.

Outline of the argument

Four-interface transmission through annuli is decomposed spectrally to separate the conformal information at each end. Uniform signed-state transfer fixes the amplitude, and an intensity derivative connects the transmission calculation to the expected pivotal count.

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

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