Why an infinite percolation cluster would disappear exactly at criticality

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Family 213 · Probability and statistical mechanics
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-11

1. The question

Take an infinite network and retain each connection independently with probability $p$. At small $p$, the remaining connected pieces are finite. At sufficiently large $p$, an infinite connected cluster may emerge. What happens at the exact boundary between those two regimes?

There are two possibilities. An infinite cluster might already exist at the critical probability, or it might appear only when the probability is increased above criticality. Knowing the threshold exists does not decide which possibility occurs.

The main claim of this family is that criticality itself has no infinite cluster for every infinite connected locally finite quasi-transitive graph whose bond threshold is below one. Quasi-transitivity means that the network has only finitely many kinds of vertices when viewed through its symmetries.

A companion gives a direct treatment of the three-dimensional cubic lattice, including both bond percolation, where edges are retained, and site percolation, where vertices are retained. The distinction matters: the general graph theorem is stated for bonds, while the site conclusion has its own argument and scope.

2. Background and essential definitions

Clusters and thresholds

Let $G=(V,E)$ be an undirected graph. In Bernoulli bond percolation, every edge is independently open with probability $p$. The open cluster $C_x$ of a vertex $x$ consists of all vertices reachable from $x$ using open edges.

Write $\mathbb P_p$ for the probability law. The critical probability is

$$p_c(G)=\inf\{p\in[0,1]:\mathbb P_p(\text{an infinite open cluster exists})>0\}.$$

The infimum specifies a boundary, not whether the boundary belongs to the percolating regime. The theorem concerns that endpoint.

For a chosen vertex $o$, define

$$\theta_o(p)=\mathbb P_p(|C_o|=\infty).$$

On a transitive graph, symmetry makes this quantity independent of the chosen vertex. On a quasi-transitive graph, several vertex types may have different probabilities, and the statement must cover all of them.

In site percolation, each vertex rather than each edge receives an independent open-or-closed bit. An open path must have every vertex open, including both endpoints. Bond and site thresholds are separate numbers even when the underlying graph is the same.

Which networks are covered?

A graph is locally finite if each vertex has finitely many neighbors. An automorphism is a relabeling of vertices preserving adjacency. A graph is transitive if any vertex can be moved to any other by an automorphism, and quasi-transitive if there are only finitely many automorphism orbits.

The lattices $\mathbb Z^d$, periodic networks with finitely many vertex types, and regular infinite trees provide examples. In the nearest-neighbor lattice, vertices are integer coordinate vectors and edges change one coordinate by one.

The condition $p_c<1$ cannot be omitted. The integer line $\mathbb Z$ has threshold one: below one, closed edges interrupt every infinite path; at one, all edges are open and the entire graph is an infinite cluster.

Critical extinction does not mean small clusters

A finite cluster can be enormous. Saying that every cluster is finite at criticality does not give a uniform bound on its size, or imply that the expected cluster size is finite. Critical systems can have connections on arbitrarily large scales without containing a genuinely infinite connected component.

3. Development history and previous partial results

Broadbent and Hammersley introduced percolation in 1957 to study transport through random media. In two dimensions, Harris and Kesten established the critical bond picture on the square lattice; Russo treated the corresponding critical site question. Planarity supplies powerful separation arguments that do not transfer directly to three dimensions.

In sufficiently high dimensions, Hara and Slade’s lace expansion established mean-field critical behavior. Fitzner and van der Hofstad extended the nearest-neighbor bond range to dimensions at least eleven. The analogous site theory has its own sufficiently-high-dimensional results; the explicit bond dimension bound should not be reused as a site bound.

Benjamini and Schramm proposed the general quasi-transitive criticality conjecture in 1996. Results beyond lattices treated substantial classes of graphs. Benjamini, Lyons, Peres, and Schramm proved critical nonpercolation under nonamenability and suitable symmetry assumptions. Hutchcroft later proved it for every quasi-transitive graph of exponential growth. M2 uses that entire established theorem and concentrates its new arguments on subexponential growth.

Subcritical sharpness theorems describe behavior strictly below $p_c$, but do not alone settle the endpoint. Similarly, critical extinction on each fixed-width slab, even together with convergence of slab thresholds, does not immediately imply extinction at the full lattice’s threshold. Both are important examples of why taking a limit of partial results requires additional control.

The manuscripts also credit earlier public developments from 2026. These include Justin Leder’s account of a formalization reporting critical bond nonpercolation on all nearest-neighbor lattices, and public expositions of connection inequalities and their site-percolation extension. M1 presents a self-contained cubic argument with this antecedent credit. M2’s broader claim concerns all quasi-transitive graphs, beyond those reported lattice conclusions.

Thus the family combines a direct treatment of a concrete three-dimensional model with a general theorem completing the stated graph-theoretic conjecture. It does not frame every included lattice case as a first announcement.

4. What the new manuscripts claim

M2: all quasi-transitive graphs in the stated class

M2 states that if $G$ is infinite, connected, locally finite, undirected, and quasi-transitive, and $p_c(G)<1$, then

$$\mathbb P_{p_c(G)}(\text{an infinite open cluster exists})=0.$$

Equivalently, $\theta_x(p_c(G))=0$ for every vertex $x$. The graph is connected and locally finite, hence countable, so the absence of an infinite cluster at each vertex excludes one anywhere in the graph.

This is a bond-percolation theorem. It includes nearest-neighbor bond percolation on $\mathbb Z^d$ for every $d\ge2$, but its main scope also covers graphs far from Euclidean lattices.

M1: both models on the cubic lattice

M1 separately states that nearest-neighbor bond percolation on $\mathbb Z^3$ has no infinite cluster at its bond critical parameter, and site percolation on $\mathbb Z^3$ has none at its site critical parameter.

For either cubic model, the order parameter $\theta(p)$ therefore tends to zero as $p$ decreases to its critical value from above. One way to understand this consequence is to approximate infinite connectivity by reaching the boundary of a large finite box. The probability of that finite event is continuous in $p$, and at criticality those probabilities tend to zero as the box grows.

Neither manuscript’s headline conclusion gives the numerical values of the cubic thresholds, the rate at which $\theta(p)$ vanishes, or the critical exponents describing cluster statistics.

5. Why the advance matters

Importance: 5/5. The general claim would settle a central conjecture about phase transitions on symmetric infinite networks. Its reach extends from familiar lattices to the much wider class of quasi-transitive graphs.

The endpoint is conceptually important. A threshold separates two regimes, but its own behavior determines whether the infinite-cluster probability starts continuously from zero. The theorem would establish that qualitative picture under minimal geometric symmetry and the necessary exclusion of $p_c=1$.

The cubic site theorem adds a related model in which the independent random variables are vertices. Its inclusion requires preserving different endpoint conventions, not merely replacing the word “edge” by “vertex.”

The rating reflects the broad graph theorem and the finite connection methods supporting it. It does not imply that quantitative critical behavior, or analogous statements for every dependent percolation model, would be settled.

6. The main idea

A finite inequality helps connections continue

Suppose a source $o$ can reach a set $A$ of possible relay vertices, and every relay has a good chance to reach a target $T$. One would like a lower bound on the chance that the source also reaches the target. The events are dependent, so multiplying probabilities without justification is not valid.

The manuscripts use a joint connection comparison of the form

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

The notation $x\leftrightarrow A$ means that an open path connects $x$ to at least one vertex in $A$. The claim applies in finite independent models, allowing unequal edge probabilities. M1 develops a hyperedge version, where an open hyperedge connects all vertices in a designated set, to handle site variables as well.

The comparison remains useful after specified bits have been fixed and the unexposed bits retain their product law. This supports an adaptive exploration: later choices can depend on earlier discoveries without pretending that the resulting large-scale success events are independent.

Build a forbidden cluster below the threshold

For the cubic lattice, assume an infinite cluster exists at $p_c$. The argument derives high connection probabilities between carefully chosen finite regions. Once all those regions and scales are fixed, continuity lets the same estimates persist at one slightly smaller parameter $p<p_c$.

Relay comparisons then join regions through fresh parts of the lattice. An exploration organized on a two-dimensional grid keeps the conditional failure probability small at each stage. A boundary-counting argument gives positive probability of infinitely many successful steps, producing an infinite open cluster below $p_c$. That contradicts the definition of the threshold.

The order of operations is essential: fix finitely many scales first, then decrease $p$. The source does not assume independence of the successful coarse regions or uniqueness of an infinite cluster in M1’s argument.

Divide the general graph problem by growth

M2 first uses Hutchcroft’s theorem for exponential-growth graphs. A graph’s growth counts the vertices in a ball of increasing graph radius.

For subexponential growth faster than every power, M2 uses a different contradiction. Assuming critical percolation, it finds large clusters at a slightly smaller parameter and brings two distinct large clusters close together. Adaptive searches and relative-entropy estimates control the cost of turning this into two large clusters across one edge. A bound on the probability of that adjacent-cluster event then rules out the construction.

In the alternative growth regime, a structural theorem supplies a quotient with controlled algebraic geometry. Roughly, it provides usable coordinates for arranging two independent directions of travel. M2 builds fresh corridors and applies the relay exploration on the original graph; the quotient organizes the routes but does not replace the percolation model.

Together these branches cover subexponential growth without requiring a complete classification of intermediate-growth graphs.

7. Scope, limitations, and open questions

Quasi-transitivity, independence, local finiteness, connectedness, and $p_c<1$ are all part of the general statement. Removing them changes the problem. In particular, arbitrary irregular infinite networks are not covered.

The general theorem is for bonds. The separate site result stated here is for the cubic lattice; it should not be promoted to a theorem about all quasi-transitive graphs by analogy.

The claims are qualitative. They do not identify critical thresholds, give universal critical exponents, or bound the tails of critical cluster sizes in every graph. Those remain distinct questions even if critical extinction is known.

The methods also distinguish finite independent comparisons from arbitrary conditioning. During exploration, the remaining variables must have the independence properties required by the comparison. Extending the conclusions to dependent spin or random-cluster models would require further arguments.

8. Sources and further reading

• Benjamini and Schramm, Percolation beyond $\mathbb Z^d$, many questions and a few answers, for the general conjecture.
• Hutchcroft, Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters, for the exponential-growth case used by M2.
• Kozma and Nitzan, A reduction of the critical nonpercolation problem to a conjectured inequality, for the finite-comparison approach.
• Leder, the public guide to lattice critical-percolation formalization, and the public site-percolation exposition, for the 2026 antecedents credited by the manuscripts.

M1 gives the cubic bond and site treatment; M2 gives the general bond theorem and its growth alternatives.

Source manuscripts

• M1. OpenAI, Critical bond and site percolation on the cubic lattice (September 24, 2026).
• M2. OpenAI, No percolation at criticality on quasi-transitive graphs (September 24, 2026).

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