Schur function at generic points and limit shape of perfect matchings on contracting square hexagon lattices with piecewise boundary conditions

We obtain a new formula to relate the value of a Schur polynomial with variables $(x_1,\dots ,x_N)$ with values of Schur polynomials at $(1,\dots ,1)$. This allows to study the limit shape of perfect matchings on a square hexagon lattice with periodic weights and piecewise boundary conditions. In particular, when the edge weights satisfy certain conditions, asymptotics of the Schur function imply that the liquid region of the model in the scaling limit has multiple connected components, while the frozen boundary consists of disjoint cloud curves.