Computing an expected hitting time for the 3-urn Ehrenfest model via electric networks

We study a three-urn version of the Ehrenfest model. This model can be viewed as a simple random walk on the graph represented by Z3M, the M-fold direct product of the cyclic group Z3 of order 3, where M is the total number of balls distributed in the three urns. We build an electric network by placing a unit resistance on each edge of the graph. We then apply a series of circuit analysis techniques, including the series and parallel circuit laws and the Delta-Y transformation, to establish shorted triangular resistor networks. A recurrence relation is derived for the effective resistance between two corner vertices of the triangular resistor networks. The recurrence relation is then used to obtain an explicit formula for the expected hitting time between two extreme states where all balls reside in one of the three urns.