# The Arctic Circle Revisited

## TL;DR

This paper analyzes limit shapes in the six-vertex model with domain wall boundary conditions using a correlation function called the emptiness formation probability, deriving new results including the Arctic Circle and Ellipses theorems.

## Contribution

It provides a closed-form expression for the emptiness formation probability and links it to a matrix model, extending the Arctic shape results to a broader class of vertex weights.

## Key findings

- Derived a determinant and integral expression for the correlation function.
- Established the connection to a Triple Penner matrix model at the free-fermion point.
- Proved the Arctic Circle and Ellipses theorems via saddle-point analysis.

## Abstract

The problem of limit shapes in the six-vertex model with domain wall boundary conditions is addressed by considering a specially tailored bulk correlation function, the emptiness formation probability. A closed expression of this correlation function is given, both in terms of certain determinant and multiple integral, which allows for a systematic treatment of the limit shapes of the model for full range of values of vertex weights. Specifically, we show that for vertex weights corresponding to the free-fermion line on the phase diagram, the emptiness formation probability is related to a one-matrix model with a triple logarithmic singularity, or Triple Penner model. The saddle-point analysis of this model leads to the Arctic Circle Theorem, and its generalization to the Arctic Ellipses, known previously from domino tilings.