  We first introduce the notion of Hamiltonian structure for a partial
difference equation. Then we construct some infinite quivers, and realize the
discrete KdV equation, the Hirota-Miwa equation and its various reductions as
the mutation relations of the corresponding cluster algebras. Finally, we show
that the log-canonical Poisson structures associated to these cluster algebras
give the Hamiltonian structures or the bihamiltonian structures of these
partial difference equations.
