 Upper critical dimension of percolation

Six is the predicted boundary for ordinary short-range percolation to have mean-field percolation exponents, with logarithmic corrections expected at the boundary. If the Fourier connectivity has leading small-momentum behavior $|k{|}^{-2}$, the triangle integral scales as ${\int }_0^{\epsilon }{r}^{d-7}dr$, finite exactly for $d>6$. This is a scaling argument; rigorous nearest-neighbor results require their own dimensional hypotheses.