The critical percolation threshold conjecture on high-dimensional lattices

The conjecture asserts that critical percolation on $\mathbb{Z}^d$ has no infinite open cluster. The unresolved nearest-neighbor cases are the intermediate dimensions, especially $3\le d\le6$; no source reports a proof or counterexample.

Known results

• The case $d=2$ is settled: $\theta(p_c)=0$.
• Lace-expansion methods prove the nearest-neighbor result for $d\ge11$ (2017), improving an earlier threshold $d\ge19$.
• The triangle condition implies $\theta(p_c)=0$ and is known in suitable high-dimensional settings.
• For sufficiently spread-out percolation, the result holds in dimensions $d>6$.

January 2024 reduction

A paper formulated a correlation-like inequality which, if proved, would imply $\theta(p_c)=0$ for every $d>1$. The authors report that they could neither prove nor disprove this inequality, so it is a reduction rather than a solution.

Current status (as of August 2026): The conjecture is settled for $d=2$ and nearest-neighbor $d\ge11$, while the cases $3\le d\le10$ remain open; no claimed proof, counterexample, or AI result was found.