Electric circuit induced by quantum walk

We consider the Szegedy walk on graphs adding infinite length tails to a finite internal graph. We assume that on these tails, the dynamics is given by the free quantum walk. We set the $ell^infty$-category initial state so that the internal graph receives time independent input from the tails, say $boldsymbol{alpha}_{in}$, at every time step. We show that the response of the Szegedy walk to the input, which is the output, say $boldsymbol{beta}_{out}$, from the internal graph to the tails in the long time limit, is drastically changed depending on the reversibility of the underlying random walk. If the underlying random walk is reversible, we have $boldsymbol{beta}_{out}=mathrm{Sz}(boldsymbol{m}_{delta E})boldsymbol{alpha}_{in}$, where the unitary matrix $mathrm{Sz}(boldsymbol{m}_{delta E})$ is the reflection matrix to the unit vector $boldsymbol{m}_{delta E}$ which is determined by the boundary of the internal graph $delta E$. Then the global dynamics so that the internal graph is regarded as one vertex recovers the local dynamics of the Szegedy walk in the long time limit. Moreover if the underlying random walk of the Szegedy walk is reversible, then we obtain that the stationary state is expressed by a linear combination of the reversible measure and the electric current on the electric circuit determined by the internal graph and the random walks reversible measure. On the other hand, if the underlying random walk is not reversible, then the unitary matrix is just a phase flip; that is, $boldsymbol{beta}_{out}=-boldsymbol{alpha}_{in}$, and the stationary state is similar to the current flow but satisfies a different type of the Kirchhoff laws.