Voronoi percolation: Cardy crossings and the near-critical window

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Family 224 · Probability and statistical mechanics
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-11

1. The question

Scatter points randomly across the plane. Give each point the region closer to it than to any other point, and color these regions independently black or white. Can a connected black path cross a prescribed rectangle or a more complicated shape?

This is Poisson–Voronoi percolation. Unlike a square or triangular lattice, its geometry is itself random, with cells of many shapes and sizes. Yet its large-scale critical behavior is expected to agree with that of familiar planar lattice models.

Family 224 asks three increasingly detailed questions. At the critical coloring probability, is a crossing probability given by Cardy’s conformal formula? If black becomes slightly more likely, do the thresholds at which many crossings appear have the same joint limit as on the triangular lattice? Finally, can the precise width of that transition window be written as a constant times a power of the microscopic scale?

The three manuscripts form a dependency chain. M3 claims the critical crossing formula. M1 takes that formula as an explicit input and claims a joint near-critical universality theorem. M2 uses the same input, together with results from M1, to determine the pivotal normalization more sharply.

2. Background and essential definitions

A random tessellation with two kinds of randomness

Let $\eta_\varepsilon$ be a homogeneous Poisson point process of intensity $\varepsilon^{-2}$. This means that a region of area $A$ contains on average $A\varepsilon^{-2}$ points, and counts in disjoint regions are independent. Typical neighboring points are a distance of order $\varepsilon$ apart.

The closed Voronoi cell of a point $v$ is

$$V_v=\{z\in\mathbb R^2:|z-v|\leq|z-w|\text{ for every }w\in\eta_\varepsilon\}.$$

Color each cell black with probability $p$, independently of the geometry and other colors. The critical value is $p=1/2$.

An annealed probability averages over both points and colors. A quenched probability first fixes the point configuration and averages only the colors. Consequently, a quenched crossing probability is a random number depending on the tessellation. Showing that this number becomes nearly deterministic is an additional statement beyond computing its average.

A quadrilateral, or quad, is a bounded Jordan domain with four distinct boundary marks $a,b,c,d$ in cyclic order. Its sides can be curved. A crossing between arcs $[a,b]$ and $[c,d]$ means a connected subset of the black cells, clipped to the closed domain, meeting both arcs. The cells come from the full-plane tessellation. Cells centered outside the domain can participate, and boundary points are not artificially wired together.

Coupling all coloring probabilities

Attach to every point an independent uniform mark $U_v\in[0,1]$, and declare its cell black when $U_v\leq p$. Increasing $p$ then only adds black cells. This monotone coupling lets us ask when each crossing first appears, while preserving dependencies between different crossing events.

A cell is color pivotal for a crossing if flipping its color alone changes the answer. Let $N_\varepsilon$ count these cells for a left-to-right crossing of the unit square at $p=1/2$, including cells whose centers lie outside the square. Define the deterministic scale

$$r_\varepsilon=\frac1{\mathbb E N_\varepsilon}.$$

The expectation includes geometry and colors. It is not a normalization chosen separately for each tessellation.

3. Development history and previous partial results

Cardy’s conformal-field-theory calculation predicted a crossing probability determined only by the conformal shape of a marked domain. Smirnov established this formula for critical site percolation on the triangular lattice. Subsequent SLE and loop-limit theory supplied a detailed continuum description there. Smirnov’s original crossing paper is the foundational rigorous example.

Voronoi percolation has exact rotational symmetry in distribution, but that alone does not determine its crossing probabilities. Bollobás and Riordan established its critical probability of one half. Tassion proved scale-uniform box-crossing estimates. These estimates keep crossing probabilities away from zero and one for fixed rectangle shapes; they do not identify their exact limits.

Ahlberg, Griffiths, Morris, and Tassion showed concentration of quenched crossing probabilities and developed quenched noise sensitivity. Vanneuville’s annealed scaling relations and quantitative quenched arm estimates supplied further control of rare connections and shared random geometry. The new crossing claim builds on this probabilistic foundation.

The near-critical mechanism was clarified by Kesten’s scaling relations: a macroscopic crossing changes when one of its rare pivotal sites changes color. Garban, Pete, and Schramm constructed triangular-lattice near-critical and dynamical scaling limits using continuum pivotal measures. M1 uses that model as its reference, with an explicit normalization through the mean pivotal count.

There is also a distinction between exponents and amplitudes. An exponent predicts a power only up to smaller logarithmic-scale errors. A limiting amplitude gives a fixed positive multiplicative constant. Sharp triangular-lattice arm asymptotics were already known in work of Du, Gao, Li, and Zhuang; M2 concerns a specific Voronoi pivotal observable under its stated crossing input.

4. What the new manuscripts claim

Cardy’s formula for the random-cell model

M3 claims that every fixed bounded Jordan quad has the annealed critical crossing limit

$$\lim_{\varepsilon\downarrow0}\mathbb P(C_\varepsilon) =F(x) =\frac{\displaystyle\int_0^x[u(1-u)]^{-2/3}\,du} {\displaystyle\int_0^1[u(1-u)]^{-2/3}\,du}.$$

Here a conformal map to the upper half-plane sends $a,c,d$ to $0,1,\infty$, and sends $b$ to $x\in(0,1)$. Thus changing the domain while preserving this conformal parameter preserves the limiting probability. For the symmetric square crossing, the value is one half.

The domain need not have a smooth boundary; the Jordan property is the stated assumption. The theorem concerns the exact closed-cell crossing event, averaged over both sources of randomness. M3 explicitly does not claim the full interface law or a quenched crossing theorem for every Jordan domain as its main result.

For a fixed rectangle, however, combining the claimed limit with established quenched concentration gives

$$\mathbb E_{\rm geom}\!\left[ \left|\mathbb P(C_\varepsilon\mid\eta_\varepsilon)-F(x)\right|^2 \right]\longrightarrow0.$$

This is a mean-square statement over tessellations, and implies convergence in probability. No coupling of tessellations at different meshes is required.

A common joint law of near-critical thresholds

For a real parameter $\lambda$, set

$$p_\varepsilon(\lambda) =\max\{0,\min\{1,\tfrac12+\lambda r_\varepsilon\}\}.$$

For a quad $Q$, let $\tau_\varepsilon(Q)$ be the first $\lambda$ at which its crossing occurs in the monotone coupling. This is random because the cell marks are random, even after fixing the geometry.

M1 fixes a countable list of all simple rational polygonal quads with rational boundary marks. It records the entire vector of their thresholds. It defines the same object for triangular-lattice site percolation, normalizing that model by its own expected unit-square pivotal count.

Assuming the scalar Cardy statement of M3, M1 claims that the triangular threshold vectors have a limiting law $\nu_\triangle$, and that

$$\operatorname{Law}\!\left( (\tau_\varepsilon(Q))_Q\mid\eta_\varepsilon \right) \longrightarrow\nu_\triangle$$

in probability over tessellations, in a metric for weak convergence of laws. The threshold coordinates are compactified by allowing $\pm\infty$, and the topology records every finite set of coordinates.

In concrete terms, choose finitely many polygonal regions and finitely many near-critical parameters. The joint distribution of all their crossing answers converges to the triangular reference under the same monotone coupling. This includes their dependencies. Equality of one crossing probability at $\lambda=0$ would not supply that information.

A sharp amplitude for pivotal cells

Under the same scalar Cardy assumption, M2 claims a constant $c_V\in(0,\infty)$ such that

$$\mathbb E N_\varepsilon\sim c_V\varepsilon^{-3/4}, \qquad r_\varepsilon\sim c_V^{-1}\varepsilon^{3/4}.$$

The symbol $\sim$ means that the ratio tends to one. The constant refers to the specified unit square, intensity convention, closed-cell event, and count of color-pivotal cells. Its positivity and existence are claimed; a numerical value is not supplied by this statement.

Why does $3/4$ appear? A bulk pivotal cell needs four alternating black and white connections from its neighborhood to macroscopic distance. The predicted four-arm probability has exponent $5/4$, while there are order $\varepsilon^{-2}$ candidate cells. Multiplying suggests $\varepsilon^{-2}\varepsilon^{5/4}=\varepsilon^{-3/4}$. This explains the exponent but not the limiting constant. M2’s additional content is that no unresolved slowly varying or oscillatory multiplier remains.

5. Why the advance matters

Importance: 5/5. The family targets a central universality question for a planar model with genuinely random geometry. The claimed crossing theorem would identify its conformal law, while the companion results would transfer substantially richer near-critical information from the triangular lattice.

The quenched conclusion is especially informative: a typical fine tessellation would have the same limiting threshold law after averaging only colors. The randomness of the cells would cease to create a separate macroscopic law.

The amplitude makes the scale explicit. Doubling spatial resolution would shrink the density window by the corresponding $3/4$ power, with a fixed model-dependent conversion factor. That factor need not match the triangular model’s factor; universality appears after each model is normalized in its own units.

The importance rating applies to this combination of claims as stated. The conditional structure of the companion theorems remains part of the result, rather than being hidden by the shared family label.

6. The main idea

M3 must recover an analytic observable from irregular geometry. On the dual triangulation, local triangles can be compared with equilateral ones by affine maps. Their displacement fields split into a conformal part and a geometric defect. Exact identities around closed cluster boundaries control these quantities, but an interface calculation requires control along open pieces.

The manuscript describes a transport construction that closes those pieces using prescribed local configurations and connecting corridors. Comparing different closures isolates endpoint contributions and a common bulk term. The resulting cancellation identifies a holomorphic limit, whose boundary values determine a conformal map to an equilateral triangle and hence Cardy’s function. This addresses the irregular embedding rather than treating rotational symmetry as a substitute for conformal invariance.

M1 then extracts joint critical information by exploring colors on a fixed geometry and applying crossing tests in the unexplored regions. To change density, it expands the effect of changing finitely many color histories. Alternating-arm estimates control the contributions of multiple nearby pivotal locations.

A local comparison replaces a microscopic color insertion by a small-square modification. It retains the signs of the resulting differences, which ordinary absolute-probability estimates could lose. Two independent color replicas on the same tessellation finally control variation over geometry and yield the quenched conclusion.

M2 sharpens the scale by studying transmission of connection information across long annuli. When four interfaces survive, the relevant difference has one dominant component. For a symmetric signed insertion, that component cancels, giving faster decay for distant geometric changes. Integrating these effects controls the derivative with respect to Poisson intensity and produces a summable error around the predicted exponent. A qualitative Cardy limit is enough to start the multiscale contraction; no rate in that input is assumed.

7. Scope, limitations, and open questions

The critical theorem averages geometry and colors in arbitrary fixed Jordan quads. The near-critical theorem is a stronger conditional-law statement, but is formulated for the specified countable rational polygonal family and monotone coloring coupling. It is not automatically a theorem in every possible topology on time-dependent interfaces.

The tessellation stays fixed as $p$ varies. This differs from moving the Poisson points or refreshing colors dynamically. Earlier dynamical-percolation work motivates the methods, but these main conclusions concern monotone thresholds.

The pivotal amplitude is annealed and model-specific. It does not assert that every sampled tessellation has exactly that pivotal count, nor that an arbitrary random planar tessellation shares the same constant. All statements concern the planar homogeneous Poisson model with independent coloring.

Questions beyond the displayed claims include quantitative rates, other random geometries, and stronger path-space descriptions of near-critical interfaces. Extending such conclusions requires additional arguments; scalar crossing convergence alone is not interchangeable with a full coupled continuum model.

8. Sources and further reading

The primary papers linked in the history section explain criticality, box crossings, quenched concentration, and the triangular near-critical reference. The Voronoi diagram offers optional geometric background; all definitions needed for the main claims are given above.

Read M3 first for the claimed scalar Cardy theorem, M1 for its transfer to joint near-critical thresholds, and M2 for the sharper pivotal amplitude. Their source numbering below follows the repository rather than this dependency order.

Source manuscripts

• M1. OpenAI, From critical crossings to quenched near-critical universality in Voronoi percolation (October 5, 2026).
• M2. OpenAI, A Pivotal Amplitude for Voronoi Percolation from Cardy’s Formula (October 5, 2026).
• M3. OpenAI, Cardy’s formula for critical Poisson–Voronoi percolation (September 23, 2026).

Sources consulted on 11 October 2026 at repository revision fd4aeeb. Download the manuscript citations.