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 α = 12 , the typical value of tricritical systems, and upper critical dimension Dcr = 6. We find that the system has upper critical dimensionDcr = 6 , while for D ≤ 4 it should undergo a first order phase transition. We comment on the latter wrong result examining the approximation we used. E-mail [email protected], 22105::PESANDO, 31890::I PESANDO In this letter we would like to show how it is possible to recover the upper critical dimension of the dimer system without using renormalization group arguments but with the use of the Ginzburg criterion. In this way we solve what could appear as a contradiction: the mean field critical exponent α = 1 2 and the upper critical dimension Dcr = 6; in fact α = 1 2 is the typical value of the mean field critical exponent of the tricritical transitions that have upper critical dimension 3. Differently from previous works ([3, 4]) we do not use renormalization group arguments. We consider the following action defined on a lattice G 1 [1] Zdimer = ∫