The partition function of the four-vertex model in inhomogeneous external field and trace statistics

TL;DR

This paper computes the partition function of an exactly solvable four-vertex model with inhomogeneous external fields, linking it to plane partitions and advancing the understanding of integrable models in combinatorics.

Contribution

It provides a determinantal formula for the partition function of the four-vertex model with inhomogeneous fields and connects it to enumerative combinatorics via plane partitions.

Findings

Partition function expressed in determinantal form

Connection established between vertex model and plane partitions

Generated functions for plane partitions with fixed diagonal sums

Abstract

The exactly solvable four-vertex model with the fixed boundary conditions in the presence of inhomogeneous linearly growing external field is considered. The partition function of the model is calculated and represented in the determinantal form. The established connection with the boxed plane partitions allows us to calculate the generating function of plane partitions with the fixed sums of their diagonals. The obtained results are another example of the connection of integrable models with the enumerative combinatorics.