Mean return time of a random walk on a cube

Question

An ant sits on one corner of a cube. Every minute it walks along one of the three edges at its current corner, choosing each with equal probability. How many minutes does it take on average to come back to the corner it started from?

Short answer

$8$ minutes. The walk on the cube's $8$ vertices is a random walk on a regular graph, so its stationary distribution is uniform and the mean return time is $1/\pi=8$ . Checking by distance from the start: from distance $1$ the expected time to reach the start is $7$ , plus the first step.

Adapted from Joshi, Denson & Downes, Quant Job Interview Questions and Answers, Q3.22, p.82; question and answer rewritten by SelfMadeQuant.com.

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