Voronoi percolation: crossings, pivotals, and near-critical laws

Source and verification

This article explains family 224 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Voronoi percolation randomizes both the lattice geometry and its colors. Family 224 starts with a critical crossing formula, extracts a sharp pivotal amplitude, and transfers to a quenched near-critical law. The latter two papers explicitly take the critical crossing formula as an input.

🏷️ Preliminaries: tessellations, quads, and pivotal cells

Place Poisson points in the plane at intensity ${\epsilon }^{-2}$. The Voronoi cell of a point consists of locations at least as close to it as to any other point. Color cells independently black with probability $p$ and white otherwise.

A quad is a bounded Jordan domain with four distinct cyclically ordered boundary marks. Its crossing event connects one designated boundary arc to the opposite one through black cells within the closed domain. Cells come from the full-plane tessellation; even a nucleus outside the quad can have a cell meeting it. There is no boundary wiring.

A cell is color-pivotal if flipping only that color changes the crossing event. Let $N_{\epsilon }$ count pivotal cells for a unit-square left-to-right crossing at $p=1/2$. Annealed expectations average geometry and colors; quenched probabilities condition on the tessellation.

🏷️ Critical crossings and the two conditional extensions

The critical companion claims Cardy’s formula for annealed crossing probabilities in every bounded Jordan quad. If a conformal map sends the four marks to $0,x,1,\infty$ in the upper half-plane, then the limit is

$$F(x)=\frac{{\int }_0^x[u(1-u){]}^{-2/3}du}{{\int }_0^1[u(1-u){]}^{-2/3}du}.$$

This is a scalar crossing limit. The theorem by itself does not state convergence of all interfaces or quenched crossing convergence in every Jordan domain. (OpenAI, 2026)

Taking precisely this scalar formula as input, the pivotal companion claims

$$\mathbb{E}{N}_{\epsilon }\sim c_V{\epsilon }^{-3/4},c_V\in (0,\infty ).$$

The constant depends on the specified crossing and intensity conventions. (OpenAI, 2026a) The near-critical companion also takes Cardy’s formula as input and claims a joint quenched threshold law matching the triangular-lattice reference, with each model normalized by its own expected unit-square pivotal count. (OpenAI, 2026c)

The family thus supplies a proposed dependency chain. These notes preserve the assumptions of each companion rather than treating an input theorem as something its recipient independently proves.

🏷️ Why pivotals set the near-critical scale

For a monotone Boolean event in finitely many independent color bits, differentiating its probability gives Russo’s formula:

$$\frac{d}{dp}{\mathbb{P}}_p(C)={\mathbb{E}}_{p}N.$$

Each derivative term asks whether the event changes when one bit switches; monotonicity makes this exactly the pivotal probability. Suitable localization passes the identity to the present crossing setup.

Therefore changing $p$ by the inverse expected pivotal count produces an order-one first-order change in crossing probability. Define

$$\begin{array}{c}{r}_{\epsilon }=\frac{1}{{\mathbb{E}}_{1/2}{N}_{\epsilon }},\\ p=\frac{1}{2}+\lambda r_{\epsilon }\end{array}$$

on bounded $\lambda$ ranges for small enough mesh. The claimed amplitude yields $r_{\epsilon }\sim c_V^{-1}{\epsilon }^{3/4}$.

The power $3/4$ has a geometric explanation. A bulk pivotal cell needs four alternating black and white arms. An arm probability with exponent $5/4$, multiplied by order ${\epsilon }^{-2}$ potential sites, predicts ${\epsilon }^{-3/4}$. This exponent calculation does not prove the finite positive amplitude.

🏷️ What the joint near-critical statement retains

Give each nucleus a uniform mark $U_v$ and declare it black whenever $U_v\le p$. This couples all densities monotonically. Each rational polygonal quad has a first crossing threshold, rescaled by $r_{\epsilon }$ around $1/2$.

The manuscript uses a fixed countable enumeration of these quads and a product topology on their threshold coordinates. Conditional on the Poisson geometry, the joint threshold law converges in probability over tessellations to the deterministic triangular reference law. This records finite collections of quads and parameters under the same monotone coupling. It is not an almost-sure coupling of all meshes or an unspecified convergence of every possible continuum configuration topology.

🏷️ The proposed proof upgrades

For the pivotal amplitude, the source compares signed differences of boundary connection states transmitted across annuli. Exactly four surviving interfaces give a leading one-dimensional contribution. A symmetry of the signed insertion cancels this leading term, leaving faster decay that is integrable over the two-dimensional location of a new Poisson point. An unsigned four-arm estimate would not provide that integrability. Differentiating in intensity and controlling the accumulated error then upgrades a power exponent to an amplitude.

For the near-critical law, the first task is to recover joint critical tests from scalar Cardy inputs using stopped color explorations and arm estimates. The second is an absolutely convergent expansion in microscopic color changes. Merging insertion locations are organized into forests of arm probes. The signed coefficients must survive comparison with mesoscopic insertions; estimates only for their absolute magnitudes would lose the threshold law’s dependence across parameters.

🏷️ Key steps and proof outlines

The three manuscripts make three distinct upgrades: a discrete cancellation becomes a holomorphic observable, a power law becomes a positive amplitude, and a critical crossing law becomes a joint quenched near-critical law. Each upgrade needs information beyond an exponent estimate.

Holomorphy with correlated geometry and colors

The critical paper decomposes the side vectors of each random triangle into a complex-linear part $U$ and an antilinear part $Q$. Their sum $V=U+Q$ is tangent displacement. An exact orthogonality identity is summed over circuit-component pairs, including isolated open-edge components; counting only geometric sides would omit terms in that identity.

The transport theorem proves that $Q-aV$ tends to zero on the relevant open arcs, for a common deterministic scalar $a$. Its proof closes arcs using local plugs, compares compatible closures, and obtains an affine displacement plus endpoint potentials. Endpoint potentials cancel when the pieces are recombined. A separate positive-area identity proves $1-2\mathrm{Re}a>0$, so $a\ne 1$. Since

$$U=(1-a)V-(Q-aV),$$

the exact identity for $U$ then forces a limiting identity for tangent displacement.

There is a further dependence problem: random edge coefficients and the color observable both depend on the same point geometry. The proof packages the coefficients into a complex measure

$${\mu }_{\epsilon }=\sum _{e=t\cap t^{\prime }}{d}_{t,e}(f(z_t)-f(z_{t^{\prime }})){\delta }_{m_e},$$

where $f$ is a compactly supported smooth test, $z_t$ is a triangle barycenter, $m_e$ an edge midpoint, and $d_{t,e}$ its oriented side vector. Midpoint integration and Stokes’ theorem show that this random measure converges against deterministic tests to $df\wedge dz$.

To multiply by the correlated observable, first tile the support into fixed squares of side $s$. On each square $B$, the total variation has $L^2$ norm at most $C{s}^2$, while an arm estimate bounds the mean oscillation indicator by $C(s/\rho {)}^c$, where $\rho$ is the fixed distance to the boundary. Cauchy–Schwarz therefore bounds the replacement error on one square by $C{s}^{2+c/2}$. Summing over $O(s^{-2})$ squares leaves $O(s^{c/2})$.

At fixed $s$, the mass ${\mu }_{\epsilon }(B)$ converges in $L^1$ to a deterministic number. It can thus be multiplied by the bounded, correlated representative of the observable on $B$. Taking the mesh limit first and then $s\downarrow 0$ gives

$${\int }_{D}Fdf\wedge dz=0.$$

This is the distributional Cauchy–Riemann equation. Continuity and local mollification make $F$ holomorphic. The three prescribed boundary arcs identify the conformal map to an equilateral triangle, and hence Cardy’s crossing value. The method never factors an expectation of geometry times colors into a product of expectations. (OpenAI, 2026b)

An integrable logarithmic derivative produces the amplitude

Write $f(R)$ for the pivotal companion’s calibrated local insertion probability at macroscopic radius $R$. The four-interface transfer analysis identifies the leading exponent $\lambda =5/4$. But $f(R)=R^{-5/4+o(1)}$ still allows a factor such as $\log R$, which would prevent convergence to a finite positive amplitude.

The decisive estimate is stronger:

$$\frac{d}{dR}\log (R^{5/4}f(R))=O(R^{-1-\alpha })\text{for some }\alpha >0.$$

Its proof differentiates Poisson intensity and uses the signed insertion symmetry to cancel the leading rank-one four-interface mode. The faster remainder is integrable over insertion locations; an unsigned four-arm bound would retain the leading mode and miss this gain.

Integrate the displayed derivative from $R$ to infinity. Because the logarithm has a finite real limit, rather than merely an upper bound, it follows that

$$R^{5/4}f(R)=a(1+O(R^{-\alpha })),0<a<\infty .$$

With mesh $\epsilon$, the site density is $m_{\epsilon }={\epsilon }^{-2}$ and the local insertion probability is $S_{\epsilon }=f(1/\epsilon )$. The source’s calibration against an auxiliary crossing gives positive constants $c,B$ with

$$\begin{array}{rl}{m}_{\epsilon }{q}_{\epsilon }{S}_{\epsilon }& ⟶c,\\ q_{\epsilon }\mathbb{E}{N}_{\epsilon }& ⟶B.\end{array}$$

Therefore $q_{\epsilon }\sim (c/a){\epsilon }^{3/4}$ and $c_V=Ba/c$. The Poisson insertion used in differentiation is an analytic device; the final $N_{\epsilon }$ still counts color-pivotal cells of the specified unwired square crossing. (OpenAI, 2026a)

Exact histories retain the dependence between densities

For times $t_1,\dots ,t_n$, one uniform color mark gives a history in $\{0,1{\}}^n$. At criticality its law is

$$P_0=\frac{1}{2}{\delta }_0+\frac{1}{2}{\delta }_1.$$

For the coupled densities $1/2+t_i{q}_{\epsilon }$, the history law is exactly $P_0+q_{\epsilon }{w}_{\mathbf{t}}$, where, for $T\ge {\max }_i|t_i|$,

$$\begin{array}{rl}{w}_{\mathbf{t}}& ={\int }_{-T}^T{\delta }_{(1_{\{s\le t_i\}}{)}_i}ds\\ & -T{\delta }_0-T{\delta }_1.\end{array}$$

This signed measure has total mass zero and variation at most $4T$. Multiplying these exact laws over sites expands a finite crossing test into signed insertion kernels. For the Poisson model, a high-probability localization indicator is multiplied into the test; the proof does not condition on that event. The Poisson factorial-moment identity then rewrites sums over distinct sites as location integrals.

A centered insertion contributes zero unless the test depends on its history. That dependence requires arms in the surrounding region. Organizing nearby insertions into a merging forest provides disjoint annuli and integrable distance factors. The resulting degree-$k$ absolute coefficient is bounded by

$$C_T^k{k}^{-(1-g/2)k},1<g<2.$$

Since $1-g/2>0$, these coefficients are summable for every fixed bounded time range. This justifies passing limits through the entire expansion, rather than only matching its first derivative.

A one-bit exchange calibrates every coefficient

To replace a microscopic insertion by a mesoscopic square, compare two independent environments. A successful annular seam must work for all four combinations of their two insides and two outsides. Four alternating connections transmit exactly one bit: whether the two black runs on the inner necklace are connected. Every tested crossing containing the trial disk depends on its interior only through this bit. Exchanging insides preserves the successful event and therefore preserves the required signed comparison.

The remaining unknown site-to-square factor is determined by one common macroscopic test. Normalize using $q_{\epsilon }=(\mathbb{E}{N}_{\epsilon }(Q_0){)}^{-1}$. Russo’s formula makes its first normalized coefficient exactly one. Comparing this coefficient with its mesoscopic version forces the factor to have one positive limit, common to both models. The same factor then calibrates every fixed-degree signed kernel.

Finally use two independent color replicas on the same tessellation. Terms sharing an inserted site between replicas carry an extra factor $q_{\epsilon }$ and have total contribution $O_T(q_{\epsilon })$. The remaining limits factor into the deterministic reference law. Thus the variance over tessellations of each conditional finite test tends to zero. Finite tests and the countable threshold coordinates give the stated quenched convergence in probability. (OpenAI, 2026c)

See Also

• Family 223 explains how crossing martingales can identify Loewner curves.
• Family 218 distinguishes quenched convergence in probability from its annealed consequence.

🏷️ Manuscripts and verification scope

• From critical crossings to quenched near-critical universality in Voronoi percolation — pinned manuscript and companion source.
• A Pivotal Amplitude for Voronoi Percolation from Cardy’s Formula — pinned manuscript and companion source.
• Cardy’s formula for critical Poisson–Voronoi percolation — pinned manuscript and companion source.

No family scope page was present at lean/docs/224.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A Pivotal Amplitude for Voronoi Percolation from Cardy’s Formula.

🐻  OpenAI 2026b. Cardy’s formula for critical Poisson–Voronoi percolation.

🐻  OpenAI 2026c. From critical crossings to quenched near-critical universality in Voronoi percolation.