Upper critical dimension

The upper critical dimension is the dimension above which the Gaussian or mean-field fixed point controls critical exponents. It is four for the ordinary Ising critical point and three for its tricritical point.

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.

At the Gaussian fixed point, the engineering dimension of the coupling of ${\varphi }^{2n}$ is $2n-(n-1)d$. It vanishes at the upper critical dimension $d_c=2n/(n-1)$.