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 Painlev equation with respect to the rapidity variable of the weights.

中文翻译：

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五顶点模型的行列式公式

我们考虑具有固定边界条件的有限方格上的五顶点模型，以使模型的配置与盒装平面分区一一对应（适合给定大小的盒子的3D杨氏图） 。非均质模型的分区函数是根据行列式给出的。对于齐次模型，可以根据汉克尔行列式给出。我们还表明，在齐次情况下，分配函数是第六个Painlev方程相对于权重的速度变量的τ-函数。