Statistical mechanics, structure of matter Exactly solved models Example

Hard hexagon model

Builds on Independent set Grand canonical ensemble

Example

Let $\Lambda$ be a finite piece of the triangular lattice, a graph in which each site has six neighbours. Let $z > 0$ be the activity. A configuration is a set of occupied sites in which no two occupied sites are neighbours, that is, an independent set of the graph. The hard hexagon model is the probability measure on these sets that gives a set $A$ the weight $z^{\abs{A}} / Z$ , with grand partition function $Z = \sum_{A \text{ independent}} z^{\abs{A}}.$ The name comes from drawing a hexagon around each occupied site: the exclusion rule says that these hexagons do not overlap.

Remarks

It is a grand canonical ensemble of particles with a hard-core repulsion and no other interaction. Baxter solved it exactly in 1980, finding the free energy in closed form in terms of $q$ -series. For small $z$ the particles are spread evenly; for large $z$ they prefer one of the three sublattices of the triangular lattice. The transition between these regimes occurs at the activity $z_c = \frac{11 + 5\sqrt{5}}{2} = \varphi^5 \approx 11.09,$ where $\varphi$ is the golden ratio. Baxter's solution uses the corner transfer matrix and identities of Rogers–Ramanujan type.

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Tag	`7MJQ2ZBQ66`
URI	`m2://7MJQ2ZBQ66`
Kind	example
Writing	W0 stub
MSC	82B20, 82B23
Instance of	Lattice model
Uses	Independent set, Grand canonical ensemble
Same as	Wikidata Q17020934 Wikipedia Hard hexagon model
Changed	3 November 2026

Hard hexagon model

Example

Remarks

Reading path

Further reading

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