A walk on a path of four vertices

Topic: Classes, recurrence, and reversibility · more questions on this topic
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A simple random walk moves on the path graph with vertices 1-2-3-4. Find the stationary distribution and the period. Does P^n converge?

Worked answer (click to show)

The degrees are (1,2,2,1), and there are three edges, so 2|E|=6. The stationary distribution of the simple random walk is \pi=(1,2,2,1)/6. The path is bipartite, with parts \{1,3\} and \{2,4\}, so every return has even length and the period is 2. The powers P^n alternate between the two parts and do not converge. The time-average occupation still converges to \pi.