Comfortable place for quantum walkers 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. At every time step, a new quantum walker penetrates into the internal from the left boundary and also some existing quantum walkers in the internal goes out to the sinks located in the left and right boundaries. 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.