Comfortable place for quantum walker on finite path

We consider the stationary state of a quantum walk on the finite path, where the sink and source are set at the left and right boundaries. The quantum coin is uniformly placed at every vertex of the path graph. We set the semi-infinite paths, namely the tails, connecting to the both boundaries, on which the dynamics of the walk are free. We also set a uniformly bounded initial state on these tails so that the internal receives inflow from the left tail and releases outflow to both tails. The square modulus of the stationary state at each vertex is regarded as the comfortability for a quantum walker to this vertex in this paper. We show the weak convergence theorem for the scaled limit distribution of the comfortability in the limit of the length of the path.