A Hopf algebra can be thought of a generalization of a group. In the same way that groups generalized the Ising model, we attempt to generalize a statistical mechanics model with a Hopf algebra. In Diaconis et. al, the Hopf square map, which combines the key operations of multiplication and comultiplication, gives rise to standard Markov chains in probability theory, like shuffling and rock breaking. However, only commutative and cocommutative Hopf algebras are used.

Inspired by their work, we use the Hopf square map on a non-commutative and non-cocommutative Hopf algebra, also known as a quantum group, to give rise to a \(q\)-deformed walk on the natural numbers. We discover a phase transition in the random walk as the deformation parameter \(q\) is varied.