# Arctic Boundaries of the Ice Model on Three-Bundle Domains

## TL;DR

This paper proves the arctic boundary phenomenon for the six-vertex model on three-bundle domains, confirming predictions and providing a rigorous mathematical justification for the tangent method heuristic.

## Contribution

It extends the arctic boundary analysis to three-bundle domains and rigorously justifies the tangent method heuristic for this class of models.

## Key findings

- Arctic boundary given by explicit algebraic curves.
- Validation of the tangent method heuristic.
- Probabilistic analysis of non-crossing path ensembles.

## Abstract

In this paper we consider the six-vertex model at ice point on an arbitrary three-bundle domain, which is a generalization of the domain-wall ice model on the square (or, equivalently, of a uniformly random alternating sign matrix). We show that this model exhibits the arctic boundary phenomenon, whose boundary is given by a union of explicit algebraic curves. This was originally predicted by Colomo-Sportiello in 2016 as one of the initial applications of a general heuristic that they introduced for locating arctic boundaries, called the (geometric) tangent method. Our proof uses a probabilistic analysis of non-crossing directed path ensembles to provide a mathematical justification of their tangent method heuristic in this case, which might be of independent interest.