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 and with one of the following boundary conditions: (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, each of which has a fixed positive length in the scaling limit, such that all the vertices along each line segment are either removed or retained. 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 $x\_{i+1}=O(\frac{x\_i}{e^{N\alpha}})$, where $\alpha>0$ is a constant independent of $N$, we show that the height fluctuations converge to a sum of independent Gaussian free fields.