On Hamiltonian Structures of Partial Difference Equations

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.