Quantum groups are a noncommutative extension of the notion of a group and first appeared in the context of quantum mechanics. Now the theory of quantum groups has further developed and has become interesting in its own right. In this work we study compact and discrete quantum groups, the latter in connection with random walks and probabilistic boundaries.
Random walks on classical groups have been extensively studied and the associated probabilistic boundaries which encode information on their asymptotic behaviour, that is, what happens after an infinite number steps, have been obtained in a number of cases. In this work we concentrate on the quantum setting where the theory is still not so clear. We compute these boundaries for particular discrete quantum groups using both a functional analytic and categorical approach. It turns out in fact that the interconnection between the two offers a very powerful tool for gaining insights into this topic.