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Math-2925 · mathematics · Probability

Near-critical Voronoi universality conditional on Cardy limits

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Assuming critical Poisson–Voronoi crossing probabilities obey Cardy’s formula in all admissible quads, do the entire near-critical crossing ensembles, scaled by their own inverse expected pivotal counts, have quenched laws converging in probability to the same continuum law as independently normalized triangular site percolation? The conditional law averages coloring marks while keeping the Poisson geometry fixed.

Short English statement

Assuming critical Poisson–Voronoi crossing probabilities obey Cardy’s formula in all admissible quads, do the entire near-critical crossing ensembles, scaled by their own inverse expected pivotal counts, have quenched laws converging in probability to the same continuum law as independently normalized triangular site percolation? The conditional law averages coloring marks while keeping the Poisson geometry fixed.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2925-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2925

Assuming critical Voronoi Cardy crossings, the authors claim a quenched universal joint limit of near-critical polygon-crossing thresholds, normalized by expected pivotal counts. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Near-critical Voronoi universality conditional on Cardy limits

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Assuming critical Voronoi Cardy crossings, the authors claim a quenched universal joint limit of near-critical polygon-crossing thresholds, normalized by expected pivotal counts.

Mathematical statement of the claimed result

Result 1.
Assume that critical Poisson--Voronoi crossing probabilities converge to Cardy's value in every bounded Jordan quadrilateral. Couple colours at all densities by independent uniform site marks, declaring a site black when its mark is at most $p$. Compare full-plane Poisson--Voronoi cells of intensity $\varepsilon^{-2}$ with triangular-lattice sites of mesh $\delta$ and their hexagonal cells. For the unit-square left-to-right crossing, let $N_e^M$ count colour-pivotal sites at $p=1/2$, with $M\in\{V,\triangle\}$ denoting the model and $e$ its mesh. Set \[ r_e^M=(\mathbb EN_e^M)^{-1},\qquad p_e^M(\lambda)=\max(0,\min(1,1/2+\lambda r_e^M)),\qquad \tau_e^M(Q)=\inf\{\lambda:C_Q\text{ occurs at }p_e^M(\lambda)\}. \] Crossings stay within the closed quad, without wiring. For a fixed enumeration $(Q_j)$ of rational polygonal quads, including the unit square, use \[ \mathcal K=[-\infty,\infty]^{\{Q_j\}},\quad d_{\mathcal K}(s,t)=\sum_{j\ge1}2^{-j}|\theta(s(Q_j))-\theta(t(Q_j))|,\quad \theta(t)=2\arctan(t)/\pi. \] Expected pivotal counts are finite and positive, and both $r_e^M\to0$. There is a probability law $\nu_{\triangle}$ on $\mathcal K$ such that \[ \operatorname{Law}(\tau_\delta^{\triangle})\Longrightarrow\nu_{\triangle},\qquad d_{\mathrm{BL}}(\operatorname{Law}(\tau_\varepsilon^V\mid\eta_\varepsilon),\nu_{\triangle}) \xrightarrow{\mathbb P}0. \] The second convergence is in Poisson-geometry probability; the conditional law averages colours while keeping all geometry fixed.

Outline of the argument

At fixed typical tessellations, color-only exploration and arm estimates turn scalar crossing limits into joint SLE and loop information. Finite witnesses recover constrained polygon crossings, after which normalized pivotal dynamics identify the monotone near-critical threshold law in probability over geometries.

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

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