The dimer partition function

We apply the Ginzburg criterion to the dimer problem and we solve the apparent contradiction of a system with mean field $\alpha =\frac{1}{2}$, the typical value of tricritical systems, and upper critical dimension $D_{cr}=6$. We find that the system has upper critical dimension $D_{cr}=6$ , while for $D\le 4$ it should undergo a first order phase transition. We comment on the latter wrong result examining the approximation we used.