Mean-field behavior in dimensions 7 through 10

Establish critical mean-field behavior for nearest-neighbor Bernoulli percolation in dimensions 7 through 10, equivalently prove the asymptotic relation \(\tau_{p_c}(x)\sim A\langle x\rangle^{2-d}\) in those dimensions.

Background

The paper studies Bernoulli bond percolation on $\mathbb Z^d$ for $d>6$, where mean-field behavior is expected to hold. The critical two-point function is expected to decay as $\langle x\rangle^{2-d}$, as it does for percolation on a tree.

The authors note that the result is known for nearest-neighbor percolation in dimensions $d\geq 11$, while dimensions 7 through 10 remain unresolved. Their main theorem provides a semi-decision procedure: if a suitable subcritical initialization can be verified, then the critical asymptotic follows in finite time.