We consider the five-vertex model on a finite square lattice with fixed boundary conditions such that the configurations of the model are in a one-to-one correspondence with the boxed plane partitions (3D Young diagrams which fit into a box of given size). The partition function of an inhomogeneous model is given in terms of a determinant. For the homogeneous model, it can be given in terms of a Hankel determinant. We also show that in the homogeneous case the partition function is a τ-function of the sixth Painlevé equation with respect to the rapidity variable of the weights.
We consider solutions of the RLL-relation with the R-matrix related to the five-vertex model. We show that in the case where the quantum space of the L-operator is infinite-dimensional and coincides with the Fock space of quantum oscillator, the solution of the RLL-relation gives a phase model with two external fields. In the case of a two-dimensional quantum space, there exist two solutions each corresponding to the five-vertex model, and their special case, corresponding to the four-vertex model. We also derive explicit expressions for quantum Hamiltonians for the inhomogeneous systems in external fields, both in the finite-and infinite-dimensional cases. Bibliography: 26 titles.
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