Co-primeness preserving higher dimensional extension of $q$-discrete Painlevé I, II equations

We construct the $q$-discrete Painlevé I and II equations and their higher order analogues by virtue of periodic cluster algebras. Using particular $k \times k$ exchange matrices, we show that the cluster algebras corresponding to $k=4$ and $5$ give the $q$-discrete Painlevé I and II equations respectively. For $k \ge 6$, we obtain higher-order discrete equations that satisfy an integrability criterion, namely, the co-primeness property.