Limit theorems for quantum walks driven by many coins

We obtain some rigorous results on limit theorems for quantum walks driven by many coins introduced by Brun et al. in the long time limit. The results imply that whether the behavior of a particle is quantum or classical depends on the three factors: the initial qubit, the number of coins M, d = [t/M], where t is time step. Our main theorem shows that we can see a transition from classical behavior to quantum one for a class of three factors.