The Benjamini–Schramm nonuniqueness conjecture

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_uSource 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 sizeSource 2. Benjamini and Schramm conjectured in 1996 that nonamenability alone forces a gap between p_c and p_uSource 3. They asked this for graphs with finitely many kinds of points under symmetry, called quasi-transitive graphsSource 4.

Many special cases were known. Benjamini and Schramm proved the gap for one-ended planar graphs of this kindSource 5. Nachmias and Peres proved it when loops in the graph are long enoughSource 6. Gaboriau proved it for graphs carrying a certain kind of harmonic functionSource 7. Hutchcroft proved a stronger form for hyperbolic graphsSource 8. For Cayley graphs of groups, others proved it for some choice of generators, not for allSource 9. Each result added a geometric condition beyond nonamenabilitySource 10.

This paper proves the conjecture for bond percolation on every such graphSource 11. There is a range of p where infinitely many infinite clusters exist at onceSource 12. It also proves Hutchcroft's stronger conjecture. A threshold defined through a connection operator lies strictly between p_c and p_uSource 13. For Cayley graphs, it holds for every finite set of generatorsSource 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 geometrySource 15. The paper reduces the problem to graphs with a balanced kind of symmetry, called unimodular. Hutchcroft's earlier work handles the other kindSource 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.