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

Abstract

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