A convergence time of Grover walk on regular graph to stationary state

We consider a quantum walk model on a finite graph which has an interaction with the outside. Here a quantum walker from the outside penetrates the graph and also a quantum walker in the graph goes out to the outside at every time step. This dynamics of the quantum walk converges to a stationary state. In this paper, we estimate the speed of the convergence to the stationary state on the $\kappa$-regular graph with the uniformly inserting of the inflow to the graph. We show that larger degree of the regular graph makes the convergence speed of this quantum walk model slower.