Abstract. The exactly solvable five-vertex model on a square lattice with fixed boundary conditions is considered. Application of the algebraic Bethe ansatz makes it possible to express the partition function and the boundary correlation functions of the nonhomogeneous model in the determinantal form. The relationship established between the homogeneous model and plane partitions helps to calculate its partition function. §1. IntroductionThe study of exactly solvable vertex models of classical statistical physics has been actual for many years [1,2]. One of the basic vertex models, the so-called six-vertex model, has been investigated intensively both for periodic and fixed boundary conditions;Quite recently, it was realized that the methods used for the investigation of integrable models can be applied efficiently to the solution of certain problems of enumerative combinatorics [9,10]. For example, the six-vertex model with domain wall boundary conditions is related to enumeration of the domino tilings of Aztec diamonds and to enumeration of alternating sign matrices [11]- [13], while the four-vertex and the phase models are related to enumeration of plane partitions (3-dimensional Young diagrams); see [14]- [19].The existence of the determinantal representation of the partition functions and boundary correlation functions is substantial in this direction.The five-vertex model is a special case of the six-vertex model with one vertex frozen out. For periodic boundary conditions, this model was used, in particular, in the study of interacting domain walls [20,21] and directed percolation [22].The five-vertex model on a square lattice is determined by five different configurations of arrows pointed both in and out of each lattice site. A statistical weight w k (k = 2, 3, 4, 5, 6) is ascribed to each admissible type of vertices (Figure 1). Representing the arrows pointing up or to the right by solid lines, one can get an alternative description of the vertices in terms of lines floating through the lattice sites. Since the bonds of a lattice may be only in two states -either with a line or without it, there is a one-toone correspondence between the admissible configurations of arrows on a lattice and the networks of lines -the nests of lattice paths.For the general case of a nonhomogeneous model, the statistical weights w k depend on the coordinates (i, j) of lattice sites. The partition function of the model is (1) Z = (config) (vertices)
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