Limit shape and height fluctuations of random perfect matchings on square-hexagon lattices

We study perfect matchings on the contracting square-hexagon lattice, constructed row by either from a of square grid or hexagonal lattice. Given 1×n periodic weights to edges, we consider probabilities dimers proportional product edge weights. show that partition function equals Schur then prove Law Large Numbers (limit shape) and Central Limit Theorem (convergence Gaussian free field) for corresponding height functions. also certain types near turning corner converge in distribution eigenvalues Unitary Ensemble, scaling limit when each segment bottom boundary grows linearly with respect dimension graph, frozen is cloud curve multiple tangent points (depending period) along horizontal segment.