Fluctuations of dimer heights on contracting square-hexagon lattices

We study perfect matchings on the square-hexagon lattice with $1\times n$ periodic edge weights such that the boundary condition is given by either (1) each remaining vertex on the bottom boundary is followed by $(m-1)$ removed vertices; (2) the bottom boundary can be divided into finitely many alternating line segments where all the vertices along each line segment are either removed or remained. In Case (1), we show that under certain homeomorphism from the liquid region to the upper half plane, the height fluctuations converge to the Gaussian free field in the upper half plane. In Case (2), when the edge weights $x_1,\ldots,x_n$ in one period satisfy the condition that $x_{i+1}=O\left(\frac{x_i}{N^{\alpha}}\right)$, where $\alpha>0$ is a constant independent of $N$, we show that the height fluctuations converge to an independent Gaussian free field in each connected component of the liquid region.