The hard hexagon model occurs within the framework of the grand canonical ensemble, where the total number of particles (the "hexagons") is allowed to vary naturally, and is fixed by a chemical potential.
In the hard hexagon model, all valid states have zero energy, and so the only important thermodynamic control variable is the ratio of chemical potential to temperature μ/(kT).
The exponential of this ratio, z = exp(μ/(kT)) is called the activity and larger values correspond roughly to denser configurations.
Solving the hard hexagon model means (roughly) finding an exact expression for κ as a function of z.
The mean density ρ is given for small z by The vertices of the lattice fall into 3 classes numbered 1, 2, and 3, given by the 3 different ways to fill space with hard hexagons.
The hard hexagon model can be defined similarly on the square and honeycomb lattices.