# Twenty-Vertex model with domain wall boundaries and domino tilings

## TL;DR

This paper explores the 20-Vertex lattice ice model with specific boundary conditions, establishing connections to domino tilings and alternating phase matrices, and generalizing known combinatorial correspondences.

## Contribution

It demonstrates equinumerous relationships between 20-Vertex model configurations and domino tilings, extending the ASM-DPP correspondence to new boundary conditions and models.

## Key findings

- Configurations are equinumerous to domino tilings of an Aztec-like square.
- Type 3 configurations conjecturally match domino tilings of a triangle.
- Reformulation in terms of Alternating Phase Matrices with roots of unity.

## Abstract

We consider the triangular lattice ice model (20-Vertex model) with four types of domain-wall type boundary conditions. In types 1 and 2, the configurations are shown to be equinumerous to the quarter-turn symmetric domino tilings of an Aztec-like holey square, with a central cross-shaped hole. The proof of this statement makes extensive use of integrability and of a connection to the 6-Vertex model. The type 3 configurations are conjectured to be in same number as domino tilings of a particular triangle. The four enumeration problems are reformulated in terms of four types of Alternating Phase Matrices with entries 0 and sixth roots of unity, subject to suitable alternation conditions. Our result is a generalization of the ASM-DPP correspondence. Several refined versions of the above correspondences are also discussed.