Notes on the Optical-Lattice Hard-Core Bose Gas of Aizenman--Lieb--Seiringer--Solovej--Yngvason

These are detailed expository notes on the staggered optical-lattice hard-core Bose gas of Aizenman, Lieb, Seiringer, Solovej and Yngvason. They restate and expand the arguments and their principal inputs in a common notation, and describe the central and orbitwise decompositions of thermodynamic-limit equilibrium states. The purpose is to make compressed or delegated steps explicit, not to introduce a new method, strengthen the estimates, or otherwise improve the cited results.