Combinatorics of a strongly coupled boson system

We introduce a quantum phase model as a limit for very strong interactions of a strongly correlated $q$-boson hopping model. We describe the general solution of the phase model and express scalar products of state vectors in determinant form. The representation of state vectors in terms of Schur functions allows obtaining a combinatorial interpretation of the scalar products in terms of nests of self-avoiding lattice paths. We show that under a special parameterization, the scalar products are equal to the generating functions of plane partitions confined in a finite box. We consider the two-dimensional vertex model related to the phase model and express the vertex model partition function with special boundary conditions in terms of the scalar product of the phase model state vectors.