We construct cluster algebras the variables and coefficients of which satisfy the discrete mKdV equation, the discrete Toda equation and other integrable bilinear equations, several of which lead to q-discrete Painlev\'e equations. These cluster algebras are obtained from quivers with an infinite number of vertices or with the mutation-period property. We will also show that a suitable transformation of quivers corresponds to a reduction of the difference equation.
Categories: math-ph math.MP