Archimedean Ice

TL;DR

This paper explores the boundary phenomena of the ice model on various Archimedean lattices, revealing both universal behaviors and unique long-range properties on the 3.4.6.4. lattice.

Contribution

It extends the study of the Arctic Circle phenomenon to new lattices and provides critical connectivity results for efficient configuration generation.

Findings

Boundary dependency extends to triangular and Kagome lattices.

Critical connectivity guarantees efficient configuration generation.

Unique long-range behavior observed on the 3.4.6.4. lattice.

Abstract

The striking boundary dependency (the Arctic Circle phenomenon) exhibited in the ice model on the square lattice extends to other planar set-ups. We present these findings for the triangular and the Kagome lattices. Critical connectivity results guarantee that ice configurations can be generated using the simplest and most efficient local actions. Height functions are utilized throughout the analysis. At the end there is a surprise in store: on the remaining Archimedean lattice for which the ice model can be defined, the 3.4.6.4. lattice, the long range behavior is completely different from the other cases.