1 · The essentials
Imagine an endless network in which each link is independently kept with the same probability. The question is whether the links that remain can connect infinitely many points. There is a critical probability marking the threshold for this possibility, and the delicate question is what happens exactly there. This family reports that no infinite connected group exists at that threshold for a broad class of networks, provided the threshold is less than one. These networks must be connected, have only finitely many links at each point, and have only finitely many types of points when viewed through the network’s symmetries. The family also establishes the conclusion for the three-dimensional cubic grid, both when links are randomly kept and when points are randomly kept instead. The cubic-grid result has a Lean formalization; the supplied source does not identify the broader network result as formalized.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, picture an endless three-dimensional arrangement of junctions, with each connecting road independently open or closed. The result says that at the precise critical opening probability, there is almost surely no connected open road system reaching infinitely many junctions. The image is imperfect because the statement concerns an infinite mathematical network rather than a physical road system.
Why it matters
Locating a threshold does not by itself settle what happens exactly at that threshold. The broader manuscript resolves the bond form of the Benjamini–Schramm criticality conjecture under the stated graph assumptions. The cubic-grid manuscript also addresses random removal of points, not just links.
2 · Going further
In Bernoulli bond percolation, each edge of a graph is independently retained with probability , and one studies the connected clusters of retained edges. The precise criticality question is whether an infinite cluster exists at , the critical probability. Manuscript 2 shows in the manuscript that almost surely no infinite cluster exists at criticality on every infinite, connected, locally finite, quasi-transitive graph with . Here, locally finite means each vertex has finitely many neighbors, and quasi-transitive means the graph’s symmetries divide its vertices into finitely many types.
The source identifies this general statement as a resolution of the bond form of the criticality conjecture of Benjamini and Schramm. Its assumptions matter: the supplied statement is restricted to bond percolation and explicitly requires . It does not state a corresponding site-percolation result for all such graphs.
Manuscript 1 addresses nearest-neighbor Bernoulli bond and site percolation on , with no infinite cluster at their respective critical parameters. Its abstract describes an argument combining a finite connection inequality for independent hyperedges, a finite-scale extension estimate, and an adaptive exploration. These are the supplied ingredients; the source gives no further detail about how they are combined. This main manuscript is marked as Lean-formalized, while the companion manuscript supplies the broader graph-level bond result without a stated formalization.
Useful notions
- Graphs, vertices, and edges
- Independent random choices
- Connected clusters
- Critical probability
- Graph symmetries
What was known before / what is new
Before
The source names the Benjamini–Schramm criticality conjecture as the context for the broader bond-percolation result. It does not describe earlier partial results or the historical status of the separate cubic-grid cases.
New
The main manuscript establishes the absence of an infinite cluster at the respective critical probabilities for nearest-neighbor bond and site percolation on , and this main result has a Lean formalization. The companion manuscript shows in the manuscript the same critical absence for bond percolation on every infinite, connected, locally finite, quasi-transitive graph with . The source describes the latter as resolving the bond form of the Benjamini–Schramm conjecture.
Limits and assumptions
The general graph result assumes an infinite, connected, locally finite, quasi-transitive graph and a critical probability strictly below one. It concerns Bernoulli bond percolation; the supplied site-percolation conclusion is specifically for nearest-neighbor percolation on the cubic lattice. “Almost surely” means with probability one, not for every possible configuration. The main cubic-lattice result has a Lean formalization, but the source does not state that the companion’s general graph result is formalized or give a finer breakdown of formalized intermediate claims. It lists no remaining open problems, and no conclusion beyond these scopes should be inferred.
Mini glossary
- Bernoulli bond percolation —
- A model in which each edge of a graph is independently retained with a common probability.
- Site percolation —
- A model in which vertices, rather than edges, are independently retained, and connectivity uses neighboring retained vertices.
- Critical probability —
- The threshold probability for the possibility of an infinite connected cluster.
- Locally finite graph —
- A graph in which every vertex has only finitely many neighbors.
- Quasi-transitive graph —
- A graph whose vertices fall into finitely many types under its symmetries.
- Almost surely —
- With probability one.
Keywords : Critical percolation · Quasi-transitive graphs · Cubic lattice
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with : at the critical probability, there is almost surely no infinite cluster. The family also establishes this conclusion for both nearest-neighbor bond and site percolation on ℤ3.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Critical bond and site percolation on the cubic lattice
24/09/2026We prove that nearest-neighbor Bernoulli bond and site percolation on ℤ3 have no infinite cluster at their respective critical parameters. The proof combines a finite connection inequality for independent hyperedges with a finite-scale extension estimate and an adaptive exploration.
No percolation at criticality on quasi-transitive graphs
24/09/2026We prove that critical Bernoulli bond percolation has no infinite cluster on any infinite connected locally finite quasi-transitive graph with critical probability less than one. This resolves the bond form of the criticality conjecture of Benjamini and Schramm.