Order by disorder up to arbitrarily high temperature

We prove that a class of classical lattice models on ${\mathbb{Z}}^d$ ($d\ge 2$) with on-site space ${\mathbb{N}}_0$ and nearest neighbour interaction, exhibits long-range checkerboard order at sufficiently high temperature. The ordering mechanism is purely entropic. The class of models contains the recently introduced model of Han–Huang–Komargodski–Lucas–Popov (arXiv:2503.22789), by which our work is inspired. The proof uses Pirogov–Sinai theory and the key input is a Peierls bound.