Triangles on ${\mathbb{Z}}^2$ and sliding phenomenon

Abstract

We confirm the list from \cite{MSS} of values $D$ for which the high-density hard-core model on ${\mathbb{Z}}^2$ with exceptional distance $D$ has infinitely many extremal Gibbs states. As a byproduct, we prove that for all $D>0$ there exists an acute-angled triangle inscribes in ${\mathbb{Z}}^2$ with side-lengths at least $D$ and area $\sqrt{3}/4\cdot D^2+O(D^{4/5})$.

6 pages; v2 - all statements made quantitative