Math Conjectures

No Infinite Cluster at Critical Percolation on Zd\mathbb Z^d

OPENLandmarkConjectureProposed c. 1960 · Canonical special case

Canonical statement

For Bernoulli bond percolation on Zd\mathbb Z^d, independently declare each nearest-neighbor edge open with probability pp, and let pc(d)p_c(d) be the threshold for an infinite open cluster. Then for every d≥2d\ge2,
Ppc(d)(0↔∞)=0. \mathbb P_{p_c(d)}(0\leftrightarrow\infty)=0.
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For Bernoulli bond percolation on \(\mathbb Z^d\), independently declare each nearest-neighbor edge open with probability \(p\), and let \(p_c(d)\) be the threshold for an infinite open cluster. Then for every \(d\ge2\),
\[
  \mathbb P_{p_c(d)}(0\leftrightarrow\infty)=0.
\]
PROB-003:[Grimmett1999Percolation][KozmaNitzan2024CriticalPercolation]
Probability theoryMathematical physics

Notes

At criticality, Bernoulli percolation is expected to have no infinite cluster on every Zd\mathbb Z^d. The two-dimensional and sufficiently high-dimensional cases are theorems, while dimensions three through six form the canonical unresolved range [Grimmett1999Percolation]. Recent work gives powerful reductions but still treats the vanishing of the critical percolation probability as open there [KozmaNitzan2024CriticalPercolation].

References (2)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.