The Six-Vertex Model with a Non-Standard Boundary Condition

TL;DR

This paper studies the six-vertex model with boundary conditions defined by integer partitions, revealing polynomial enumeration of states and providing explicit counts for specific shapes like hooks and staircases.

Contribution

It introduces a method to enumerate states of the six-vertex model with boundary conditions based on integer partitions, including explicit formulas for certain shapes.

Findings

Number of states is a polynomial in the largest part of the partition.

Complete enumeration for hook shapes and staircases.

Provides a new combinatorial approach to boundary conditions in the six-vertex model.

Abstract

We consider the enumeration of states in the Brubaker-Bump-Friedberg six-vertex model, whose boundary conditions are determined by an integer partition. In general, we find the number of states is a polynomial in the largest part of the partition. By explicating this technique, we also enumerate the states completely for hook shapes and staircases.