A simple model of quantum walk with a gap in distribution

Quantum walks are quantum counterparts of random walks, and their probability distributions are different from each other. A quantum walker distributes on a Hilbert space, and it is observed at a location with a probability. The finding probabilities have been investigated, and some interesting things have been analytically discovered. They are, for instance, ballistic behavior, localization, or a gap. We study a 1-dimensional quantum walk in this paper. Although the walker launches off at a location under a localized initial state, some numerical experiments show that the quantum walker does not seem to distribute around the launching location, which suggests that the probability distribution holds a gap around the launching location. To prove the gap analytically, we derive a long-time limit distribution, from which one can tell more details about the finding probability.