Critical percolation and finite connection inequalities

Source and verification

This article explains family 213 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

At the critical density of independent percolation, is there already an infinite cluster? Family 213 claims a negative answer for bond percolation on quasi-transitive graphs with critical probability below one, together with bond and site results on cubic lattices. Its arguments turn highly probable finite connections into a contradiction below criticality.

🏷️ Preliminaries: criticality and graph symmetry

In bond percolation each edge is independently open with probability $p$. In site percolation vertices are open instead. An open cluster is a connected component of the open configuration. For a fixed vertex $o$, write

$$\begin{array}{rl}{\theta }_o(p)& ={\mathbb{P}}_p(o\leftrightarrow \infty ),\\ p_c& =\inf \{p:{\theta }_o(p)>0\}.\end{array}$$

On a connected graph the threshold does not depend on the starting vertex. A graph is locally finite if each vertex has finite degree, and quasi-transitive if its symmetries have only finitely many vertex orbits. This allows several local vertex types while retaining uniform large-scale control.

The question concerns the value at $p_c$. The definition already rules out percolation below it and says nothing by itself about the endpoint.

🏷️ What is claimed

Critical absence, as stated in the companions

On every infinite connected locally finite quasi-transitive graph with $p_c<1$, independent bond percolation has no infinite cluster at $p_c$. (OpenAI, 2026)

On ${\mathbb{Z}}^d$, $d\ge 2$, the lattice companion claims the corresponding result for both bond and site percolation, each at its own critical parameter. (OpenAI, 2026a)

The assumption $p_c<1$ matters: at parameter one every edge is open. The site statement is not obtained merely by replacing the word “edge” with “vertex” in the graph theorem. The finite product-space formulation must encode the different local connection rules.

🏷️ An elementary consequence: continuity from above

Let $B_n$ exhaust the graph, and let $A_n$ be the event that $o$ connects to its boundary. These events decrease to $\{o\leftrightarrow \infty \}$. If ${\theta }_o(p_c)=0$, choose $n$ with ${\mathbb{P}}_{p_c}(A_n)<\epsilon$. The event uses finitely many bits, so its probability is a polynomial in $p$ and remains below $2\epsilon$ for $p>p_c$ sufficiently close to $p_c$. Since ${\theta }_o(p)\le {\mathbb{P}}_p(A_n)$, this proves right-continuity at criticality.

This short deduction explains the article’s connection with continuity of the phase transition. It does not establish the critical absence on which it depends.

🏷️ The finite gluing estimate

A central finite inequality has the form

$$\begin{array}{rl}& \mathbb{P}(o\leftrightarrow A,o\leftrightarrow T)\\ & \ge \mathbb{P}(o\leftrightarrow A)\underset{a\in A}{\min }\mathbb{P}(a\leftrightarrow T).\end{array}$$

Here connections are in the specified finite independent connection model; the source also treats its hyperedge version. If every possible relay point $a$ reaches $T$ with probability at least $1-b$, then reaching the relay set but failing to reach $T$ has probability at most $b\mathbb{P}(o\leftrightarrow A)$.

The events are correlated. The estimate is useful precisely because it does not pretend that discovering the relay point leaves an independent connection event. Its conditional versions retain a product law on the unexposed bits after already inspected bits have been fixed.

🏷️ From finite relays to a subcritical contradiction

Assume an infinite critical cluster exists. The lattice argument extracts large finite seeds and connections to selected boundary portions with probabilities close to one. Since only finitely many bits are involved in each estimate, the estimates persist at a slightly smaller parameter. A controlled collection of relay moves then explores an auxiliary two-dimensional lattice.

At each stage, the exploration exposes only the bits needed for that move. The conditional gluing inequality keeps its failure probability small even though the coarse moves are dependent. A boundary-counting argument shows that enough successful moves continue indefinitely. This creates an infinite cluster at a parameter below $p_c$, contradicting its definition. Checking the exposure rule is essential: high marginal success probabilities alone do not imply coarse percolation.

The general graph companion separates growth regimes. It uses an existing critical-absence input in the exponential-growth regime. In subexponential growth it combines estimates for neighboring large clusters with controlled searches; graphs having polynomial growth along a sequence require a structural reduction that supplies two usable directions and connecting corridors. The geometric construction replaces the coordinate boxes available on ${\mathbb{Z}}^d$, while the conditional finite gluing serves the same purpose.

🏷️ Key steps and proof outlines

The decisive invariant is a conditional promise for each unexplored box. It is stronger than a large unconditional crossing probability and is preserved by a carefully restricted exposure rule. The finite comparison supplies the promise; an adaptive contour argument turns it into an infinite cluster below criticality (OpenAI, 2026a).

Finite gluing through positive connection weights

Here is the mechanism behind the finite inequality, without an independence assumption on a discovered relay. Order the relay vertices $x_1,\dots ,x_{n-1}$ and put the source $x_n=o$ last. For each $k$, condition the cluster of $x_k$ to avoid all earlier listed vertices. Let $h_k(i)$ be the conditional probability that this cluster contains $x_i$. The matrix $H=(h_1,\dots ,h_n)$ is unit lower triangular.

The source proves a stronger positivity lemma: if $F$ is a nonnegative increasing function of that conditioned cluster, its weighted membership column, minus its mean times $h_k$, is a nonnegative combination of later columns. The proof alternately resamples the source cluster and the clusters it must avoid. Exact-cluster conditioning leaves all hyperedges disjoint from the revealed cluster independent. Enlarging a deleted set gives a monotone nonnegative residual. Descending induction expresses this residual using later columns.

To pass from the iteration to its limit, there is a fixed positive chance of isolating the source at every resampling step. Consequently the oscillation of a function contracts by a factor less than one, and the iterated function converges uniformly to its stationary mean. This closes the positivity argument even at states having zero probability under the stationary distribution.

Applied to connection indicators, the lemma gives $H^{-1}=I-\Gamma$, where $\Gamma$ has nonnegative entries strictly below the diagonal. In the source row these entries have sum $\mathbb{P}(o\leftrightarrow A)$. The joint connection probability is therefore at least a nonnegative weighted sum of relay-to-target probabilities, proving

$$\mathbb{P}(o\leftrightarrow A,o\leftrightarrow T)\ge \mathbb{P}(o\leftrightarrow A)\underset{a\in A}{\min }\mathbb{P}(a\leftrightarrow T).$$

Parameters equal to zero or one follow by continuity of finite event probabilities. This endpoint extension is exactly what permits already exposed bits to be fixed.

The site model needs an incidence construction

For site percolation, create one starred node for every vertex and one connector node for every edge. The hyperedge at vertex $x$ consists of $x^{\ast }$ and its incident connectors, and is open precisely when site $x$ is open. Distinct starred vertices connect exactly when the corresponding sites have an open site path.

There is one discrepancy: a hypergraph node always connects to itself, whereas site self-connection requires the site to be open. On the failure event $o\leftrightarrow A$, $o\nleftrightarrow T$, the origin is already open. Its relevant connections therefore agree with those in the incidence model, including self-connection. This event inclusion transfers the failure bound. It is why simply identifying the two models at every event would be incorrect.

A promise that survives the exploration

Assuming critical percolation, the finite geometric estimates supply a scale where seeds connect to prescribed boundary portions with very high probability. Finite-event continuity preserves these estimates at one parameter $q<p_c$. Place disjoint coarse cubes indexed by ${\mathbb{Z}}^2$, and maintain a queue with one parent assigned to each discovered cube.

The source first conditions on a positive-probability finite initialization event $I$, which makes the root cube good. Throughout this exploration and the contour estimate below, probabilities and expectations are under ${\mathbb{P}}_I={\mathbb{P}}_q(\cdot \mid I)$.

For a pending cube $v$, let $H_v$ mean that the origin connects to its inner box using only its parent chain. Maintain

$${\mathbb{P}}_I(H_v\mid \mathcal{F})>1-\delta ,$$

where $\mathcal{F}$ is the revealed history. Processing $v$ exposes only its internal bits and incoming interface. For an unvisited neighbor $x$, compute the conditional child connection probability $b_x$ without exposing the child’s bits. Declare $v$ good only if $H_v$ occurs and every such $b_x>1-\delta$.

Other processing steps cannot disturb a pending child’s promise: its internal bits and incoming parent interface remain untouched. Conditional on every finite history, all unexposed bits retain their original product law. The queue and the computed probabilities depend only on already revealed values and impose no extra condition on fresh bits.

Apply the relay estimate before exposing $v$. With geometric error parameter $\eta$, it gives

$$\mathbb{E}[1_{H_v}(1-b_x)\mid \mathcal{F}]\le 13\eta .$$

Markov’s inequality bounds the probability of $H_v$ occurring but $b_x\le 1-\delta$ by $13\eta /\delta$. There are at most four children, so

$$\mathbb{P}(v\text{ bad}\mid \mathcal{F})\le \delta +52\eta /\delta <f.$$

Choose the tolerances in this order so $f$ is sufficiently small. Independence between different child tests is unnecessary.

Contours work with conditional failure bounds

For any deterministic set of $k$ coarse vertices, reveal its members in their actual processing order. Successive conditioning gives probability at most $f^k$ that all are processed and bad. The argument includes the possibility that some are never processed.

If the good cluster is finite, its boundary has a dual contour of length $n$. There are at most $2n{3}^n$ candidate contours. A matching of crossed edges supplies at least $n/7$ disjoint pairs of endpoints; for each pair, at least one endpoint must be bad. A union bound over the two endpoint choices gives contour probability at most $(2f{)}^{n/7}$. Hence

$$\mathbb{P}(\text{finite good cluster})\le \sum _{n\ge 1}2n{3}^n(2f{)}^{n/7}<1.$$

Infinitely many good cubes provide actual connections to infinitely many disjoint inner boxes, producing an infinite cluster with positive conditional probability. Multiplying this probability by ${\mathbb{P}}_q(I)>0$ gives positive unconditional probability at $q<p_c$, the contradiction.

For the quasi-transitive companion, this lattice exploration is replaced by a growth-dependent argument. Exponential growth uses the stated existing critical-absence theorem; subexponential growth splits into the source’s superpolynomial case and a polynomial-growth subsequence case. The latter invokes a structural theorem to obtain usable directions and corridors. These are separate geometric inputs, not consequences of the finite inequality alone (OpenAI, 2026b).

See Also

• The earlier percolation article discusses an Anthropic formal artifact and its finite connection inequality. That provenance is distinct from this OpenAI manuscript family.
• Family 214 studies the additional interval between existence and uniqueness on nonamenable graphs.

🏷️ Manuscripts and verification scope

• Critical bond and site percolation on the cubic lattice — pinned manuscript and companion source.
• No percolation at criticality on quasi-transitive graphs — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope page describes both the cubic-lattice bond and site claims and the general quasi-transitive bond claim. 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. Critical bond and site percolation on the cubic lattice.

🐻  OpenAI 2026b. No percolation at criticality on quasi-transitive graphs.