1 · The essentials
Imagine an infinite network in which each connection is independently kept with the same probability. The question is whether the surviving connections form just one endlessly connected piece or many separate endless pieces. For the networks covered by this result, there is a whole nonempty interval of probabilities where infinitely many such pieces coexist with probability one. This applies to connected networks with finitely many connections at each point, a particular lack of narrow boundaries, and only finitely many types of points under the network’s symmetries. The result resolves the Benjamini–Schramm nonuniqueness conjecture under these assumptions. The main result has a Lean formalization, while the supplied abstract also reports a stronger mathematical bound.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, picture an endless road network where each road independently stays open or closes. At some opening probabilities, infinitely many separate regions can each extend forever, even though no open route connects one region to another. The image is only useful for networks satisfying the result’s specific structural assumptions.
Why it matters
The result separates the appearance of infinite connected pieces from the point at which a unique infinite piece takes over. It establishes that, for every graph in the stated class, a genuine interval of nonuniqueness lies between these thresholds. The repository summary also reports that a stronger operator bound establishes the critical triangle condition.
2 · Going further
Bernoulli bond percolation retains each edge of a graph independently with probability . The precise setting here is an infinite, connected, locally finite, nonamenable, quasi-transitive graph. Writing for the threshold for the existence of an infinite cluster and for the uniqueness threshold, the main result is . The manuscript states that there is a nonempty interval of parameters on which infinitely many infinite clusters exist almost surely.
The supplied abstract identifies this as the Benjamini–Schramm bond-percolation nonuniqueness conjecture. It also reports the stronger inequality , resolving what it calls Hutchcroft’s operator-threshold conjecture. This bound places an operator threshold strictly above the existence threshold and no higher than the uniqueness threshold, thereby giving the desired strict separation.
The abstract does not describe the argument establishing that operator bound, so the supplied material supports an explanation of the implication, not a reconstruction of the method. The repository summary additionally attributes the critical triangle condition to a stronger operator bound. Only the main manuscript is supplied; no companion manuscripts or relationships between them are described.
Useful notions
- Graphs as points joined by edges
- Independent random choices
- Connected components and infinite clusters
- Probability thresholds
- Symmetries of a graph
What was known before / what is new
Before
The source identifies the strict separation of the existence and uniqueness thresholds as a conjecture of Benjamini and Schramm. It also identifies a stronger operator-threshold conjecture of Hutchcroft. No earlier partial results or historical chronology are provided.
New
The main result establishes throughout the stated class of graphs, giving a nonempty interval with infinitely many infinite clusters almost surely. The manuscript also shows the stronger inequality . The repository summary further reports the critical triangle condition as a consequence of a stronger operator bound.
Limits and assumptions
The hypotheses are essential to the stated scope: the graph must be infinite, connected, locally finite, nonamenable, and quasi-transitive, and the model is Bernoulli bond percolation. The supplied claims do not specify the behavior at the endpoints of the nonuniqueness interval. The main result has a Lean formalization, but the supplied material does not detail its coverage of the stronger operator bound or the critical triangle condition; these additional statements are reported by the manuscript and repository summary. The supplied text lists no remaining open questions and gives no proof-method details.
Mini glossary
- Bernoulli bond percolation —
- A random graph model in which each edge is independently retained with the same probability.
- Infinite cluster —
- A connected component of retained edges containing infinitely many vertices.
- Locally finite —
- Each vertex has only finitely many incident edges.
- Quasi-transitive —
- The graph’s symmetries divide its vertices into only finitely many types.
- Nonamenable —
- A graph property expressing that finite sets cannot have boundaries arbitrarily small relative to their size.
- Almost surely —
- An event occurs almost surely when it has probability one.
Keywords : Percolation · Nonuniqueness · Nonamenable graphs
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)
Proves for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition.
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
Nonuniqueness of percolation on nonamenable quasi-transitive graphs
24/09/2026We prove the bond-percolation nonuniqueness conjecture of Benjamini and Schramm: every infinite connected, locally finite, nonamenable quasi-transitive graph has a nonempty interval of parameters for which Bernoulli bond percolation almost surely has infinitely many infinite clusters. We also prove Hutchcroft's stronger operator-threshold conjecture, establishing .