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 qdiscrete Painlevé 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.

In this article we formulate a group of birational transformations which is isomorphic to an extended affine Weyl group of type (A 2n+1 + A 1 + A 1 ) (1) with the aid of mutations and permutations to a mutation-periodic quiver on a torus. This group provides four types of generalizations of Jimbo-Sakai's q-Painlevé VI equation as translations of the extended affine Weyl group. Then the known three systems are obtained again; the q-Garnier system, a similarity reduction of the lattice q-UC hierarchy and a similarity reduction of the q-Drinfeld-Sokolov hierarchy.

In this article we formulate a group of birational transformations that is isomorphic to an extended affine Weyl group of type $(A_{2n+1}+A_1+A_1)^{(1)}$ with the aid of mutations and permutations of vertices to a mutation-periodic quiver on a torus. This group provides a class of higher order generalizations of Jimbo–Sakai’s $q$-Painlevé VI equation as translations on a root lattice. Then the known three systems are obtained again: the $q$-Garnier system, a similarity reduction of the lattice $q$-UC hierarchy, and a similarity reduction of the $q$-Drinfeld–Sokolov hierarchy.

We construct the Laurent phenomenon algebras the cluster variables of which satisfy the discrete BKP equation and other difference equations obtained by its reduction. These Laurent phenomenon algebras are constructed from seeds with a generalization of mutation-period property. We show that a reduction of a seed corresponds to a reduction of a difference equation.

We introduce a class of recursions defined over the d-dimensional integer lattice. The discrete equations we study are interpreted as higher dimensional extensions to the discrete Toda lattice equation. We shall prove that the equations satisfy the coprimeness property, which is one of integrability detectors analogous to the singularity confinement test. While the degree of their iterates grows exponentially, their singularities exhibit a nature similar to that of integrable systems in terms of the coprimeness property. We also prove that the equations can be expressed as mutations of a seed in the sense of the Laurent phenomenon algebra.