Dimer model

Remarks

When $G$ is the grid graph of a region of unit squares, a perfect matching is the same as a domino tiling of the region. Kasteleyn, and independently Temperley and Fisher, found in 1961 that the number of domino tilings of an $m \times n$ rectangle is $$\prod_{j=1}^{\ceil{m/2}} \prod_{k=1}^{\ceil{n/2}} \left( 4 \cos^2 \frac{\pi j}{m + 1} + 4 \cos^2 \frac{\pi k}{n + 1} \right).$$ Check for $m = n = 2$: the only factor is $4 \cdot \frac14 + 4 \cdot \frac14 = 2$, and a $2 \times 2$ square has exactly $2$ domino tilings. Check for $m = 2$, $n = 3$: the factors are $1 + 2 = 3$ and $1 + 0 = 1$, giving the $3$ tilings of a $2 \times 3$ rectangle. For an $m \times n$ rectangle with $m, n \to \infty$, $\frac{1}{mn} \log(\text{number of tilings}) \to G_{\mathrm{Cat}}/\pi \approx 0.2916$, where $G_{\mathrm{Cat}}$ is Catalan's constant. On planar graphs $Z$ is a Pfaffian, by the FKT algorithm, which makes the model exactly solvable.