# The Benjamini–Schramm nonuniqueness conjecture > Percolation on every fast-spreading symmetric graph has a range with infinitely many endless clusters. Plain-English explainer, Lean status and sources. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-214/ Last updated: 2026-10-14 **Percolation on every fast-spreading symmetric graph has a range with infinitely many endless clusters.** **Trust:** Formal proof listed · **Papers:** 1 · **First paper:** September 24, 2026 · **Field:** Probability and statistical mechanics · **Entry:** 214 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. Open each link of an infinite network at random with chance p. On fast-spreading networks, endless clusters may appear before they merge into one. Benjamini and Schramm conjectured in 1996 that a gap between these two thresholds always exists. This paper proves it for bond percolation, with a Lean proof. It also proves a stronger operator form conjectured by Hutchcroft. ## At a glance - **First step:** 1996 (Itai Benjamini and Oded Schramm, the nonuniqueness conjecture) - **Significance:** Landmark. It settles the bond form of a central conjecture of percolation beyond the flat grid, with a Lean proof. - **Trust:** Formal proof listed - **Field:** Probability and statistical mechanics (1 paper) ## The explanation Take an infinite tree where every point has three links. Keep each link open with chance p, independently. This is bond percolation. Follow open links away from a point, one level at a time. Each point has two links leading further out. With p = 3/5, each point has on average 6/5 open links outward. A branch that grows this way survives forever with chance 5/9. So infinite open clusters appear. On this tree, they appear once p passes 1/2. But there is never just one infinite cluster, for any p below 1. Every link of a tree is the only bridge between two halves. Each closed link cuts clusters apart, so infinitely many infinite clusters coexist. A tree has two separate thresholds: one where infinite clusters appear, called p_c, and one where they merge into a single cluster, called p_u[1]. On a flat grid, the two thresholds coincide. Trees spread out very fast. Graphs with this property are called nonamenable: every finite set of points has a boundary at least a fixed fraction of its size[2]. Benjamini and Schramm conjectured in 1996 that nonamenability alone forces a gap between p_c and p_u[3]. They asked this for graphs with finitely many kinds of points under symmetry, called quasi-transitive graphs[4]. Many special cases were known. Benjamini and Schramm proved the gap for one-ended planar graphs of this kind[5]. Nachmias and Peres proved it when loops in the graph are long enough[6]. Gaboriau proved it for graphs carrying a certain kind of harmonic function[7]. Hutchcroft proved a stronger form for hyperbolic graphs[8]. For Cayley graphs of groups, others proved it for some choice of generators, not for all[9]. Each result added a geometric condition beyond nonamenability[10]. This paper proves the conjecture for bond percolation on every such graph[11]. There is a range of p where infinitely many infinite clusters exist at once[12]. It also proves Hutchcroft's stronger conjecture. A threshold defined through a connection operator lies strictly between p_c and p_u[13]. For Cayley graphs, it holds for every finite set of generators[14]. The single obstacle is that the tree's separation comes from its tree structure. The task was to get an equally robust separation from nonamenability alone, with no tree and no extra geometry[15]. The paper reduces the problem to graphs with a balanced kind of symmetry, called unimodular. Hutchcroft's earlier work handles the other kind[16]. How sure is this? A computer proof checker called Lean has a formal proof of the main statements. These include the strict gap, the nonempty range with infinitely many infinite clusters, and the Cayley graph case. Lean also covers estimates at the critical point. Each is matched to a reference statement by a tool called Comparator. No specialist review of the paper has been published. The paper covers bond percolation; the site version is not included. **What it doesn't show.** This does not cover site percolation, or give the values of the thresholds. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Landmark if it holds. The main statements have Lean proofs matched by Comparator. No specialist review is published, and only bond percolation is covered. ## Words used - **Bond percolation:** Keeping each link of a network open with chance p, independently. - **Infinite cluster:** An endless connected set of points joined by open links. - **p_c and p_u:** The chance at which infinite clusters first appear, and the chance at which there is exactly one. - **Nonamenable graph:** A graph where every finite set has a boundary at least a fixed fraction of its size. - **Quasi-transitive graph:** A graph with only finitely many kinds of points up to its symmetries. - **Cayley graph:** A graph drawn from a group and a set of generators, joining each element to its products with generators. - **Unimodular:** A balance condition on how a graph's symmetries move points around. ## The original paper - [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf) - September 24, 2026 - [Lean notes](https://github.com/openai/math/blob/main/lean/docs/214.md) ## Quoted lines from the paper (16) [1] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > A regular tree of degree at least three supplies the basic example of separated thresholds. [2] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > The graph is \emph{nonamenable} if $h_V(G)>0$. [3] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > Benjamini and Schramm conjectured that this mechanism always produces a nonempty nonuniqueness phase when the graph has finitely many vertex orbits. [4] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > It is \emph{quasi-transitive} if its automorphism group $\Aut(G)$ has finitely many orbits on $V$. [5] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > Benjamini and Schramm proved $0<\pc<\pu<1$ for one-ended nonamenable transitive planar graphs, for both bond and site percolation. [6] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > Nachmias and Peres proved nonuniqueness and triangle estimates for nonamenable transitive graphs of sufficiently large girth relative to their spectral radius. [7] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > Gaboriau proved $\pc<\pu$ for unimodular transitive graphs admitting a nonconstant harmonic Dirichlet function, using $\ell^2$ Betti numbers, and extended the argument to quasi-transitive graphs. [8] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > Hutchcroft introduced and developed the operator-threshold framework used here, proved $\pc<\ptwo$ for nonamenable Gromov hyperbolic quasi-transitive graphs, and proved it for graphs admitting a quasi-transitive nonunimodular automorphism group. [9] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > These results permit a choice of generators; … treats an arbitrary prescribed generating set. [10] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > These results explain the usefulness of geometric separation and random-walk decay, while imposing conditions beyond nonamenability. [11] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > The theorem resolves the bond-percolation nonuniqueness conjecture of Benjamini and Schramm positively, and proves Hutchcroft's stronger operator-threshold conjecture in the same graph class. [12] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > There exist $\pc The theorem resolves the bond-percolation nonuniqueness conjecture of Benjamini and Schramm positively, and proves Hutchcroft's stronger operator-threshold conjecture in the same graph class. [14] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > Let $\Lambda$ be a nonamenable finitely generated group, and let $S$ be any finite symmetric generating set with the identity removed. [15] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > A regular tree of degree at least three supplies the basic example of separated thresholds. The difficulty is to extract an equally robust separation from nonamenability alone, without a tree structure or an additional geometric hypothesis. [16] From [Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf): > The nonunimodular theorem is used directly in our reduction. ## Other sources - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” ## Related - [Seymour’s second-neighborhood conjecture](https://closedform.nishachay.tech/e/openai-math-2026-173/) - Combinatorics - [Counterexamples to Sidorenko’s conjecture and the forcing conjecture](https://closedform.nishachay.tech/e/openai-math-2026-161/) - Combinatorics - [Critical percolation on every quasi-transitive graph](https://closedform.nishachay.tech/e/openai-math-2026-213/) - Probability and statistical mechanics - [GOE bulk universality for regular graphs with weak Anderson disorder](https://closedform.nishachay.tech/e/openai-math-2026-219/) - Probability and statistical mechanics - [The Erdős–Gallai cycle-decomposition conjecture](https://closedform.nishachay.tech/e/openai-math-2026-181/) - Combinatorics - [Probability and statistical mechanics](https://closedform.nishachay.tech/science/mathematics/probability-and-statistical-mechanics/) - Field · 29 results - [Named conjectures](https://closedform.nishachay.tech/topic/named-conjectures/) - Topic · 85 results - [Graph theory](https://closedform.nishachay.tech/topic/graph-theory/) - Topic · 40 results - [Open since the 1990s](https://closedform.nishachay.tech/open-since/1990s/) - Era · 55 results - [Lean-checked results](https://closedform.nishachay.tech/trust/lean-checked/) - Trust · 242 results - [Landmark results](https://closedform.nishachay.tech/significance/landmark/) - Significance · 49 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20214%5D%20)