Percolation phase-transition puzzle
For every infinite, locally finite, transitive graph , let denote Bernoulli bond percolation with parameter , let , and define . Prove that .
References
Primary source
Additional references
Progress summary
A new paper claims to settle the long-standing question above the percolation threshold, but the exact-threshold case remains open and the proof has not been independently checked.
Benjamini and Schramm formulated the transitive-graph version in 1996. The problem asks how clusters behave below, at, and above the critical probability.
Known results
- Below the threshold, clusters are small and widely separated (Antunović and Veselić, 2007).
August 31, 2026 claimed supercritical proof
Diskin, Easo, Radhakrishnan, Sudakov, and Tassion claim that for every infinite transitive graph and , the probability that the origin lies in a large finite cluster decays exponentially in the isoperimetric function. Their arXiv paper and the accompanying report describe this as the missing supercritical half, using an improved sprinkling argument; the claim remains unverified. The separate question at remains open in general, including for the full three-dimensional grid.
Current status (as of August 2026): the supercritical half is claimed solved but unverified; the critical-threshold question remains open.
Solutions 0
No solutions have been posted yet.