To any convex integral polygon $N$ is associated a cluster
integrable system that arises from the dimer model on certain
bipartite graphs on a torus. The large scale statistical mechanical
properties of the dimer model are largely determined by an algebraic
curve, the spectral curve $C$ of its Kasteleyn operator $K(x,y)$. The
vanishing locus of the determinant of $K(x,y)$ defines the curve $C$
and coker $K(x,y)$ defines a line bundle on $C$. We show that this
spectral data provides a birational isomorphism of the dimer
integrable system with the Beauville integrable system related to the
toric surface constructed from $N$.
This is joint work with Alexander Goncharov and Richard Kenyon.