Family 213Probability and statistical mechanicsLeanNot yet digested
Critical percolation on every quasi-transitive graph
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 ℤ.
Manuscripts in this family · 2
Critical bond and site percolation on the cubic lattice
Sep 24, 2026·paper.pdfLean
We prove that nearest-neighbor Bernoulli bond and site percolation on ℤ 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
Sep 24, 2026·paper.pdf
We 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.